Thesis Title

Some Results For Fourth Order Semilinear Elliptic PDEs

A Thesis

Submitted to the Tata Institute of Fundamental Research, Mumbai for the Degree of Doctor of Philosophy in Mathematics

by

Abhishek Sarkar

Centre for applicable Mathematics Tata Institute of Fundamental Research Bangalore November, 2015

Declaration of Authorship This thesis is a presentation of my original research work. Whenever contributions of others are involved, every effort is made to indicate this clearly, with due reference to the literature and acknowledgement of collaborative research and discussions. The work was done under the guidance of Prof. Prashanth K. Srinivasan at TIFR Centre for applicable Mathematics, Bangalore.

Abhishek Sarkar In my capacity as supervisor of the candidates thesis, I certify that the above statements are true to the best of my knowledge.

Prof. Prashanth K. Srinivasan Date:

iii

Dedicated To my Maa, Baba & Didi

v

Abstract

This thesis deals with partial differential equations involving the polyharmonic operator and a semilinear term. In particular, we study problems that arise naturally while trying to prescribe the so-called Q-curvature function on the standard N -sphere, N ≥ 4 and existence/non-existence of point singularities in domains in IRN , N ≥ 4 .

The first part of the thesis deals with the prescribed Q-curvature problem wherein : (i) in Chapter 2 we prove the existence and multiplicity of solutions in standard 4-sphere and (ii) in Chapter 3 we show local uniqueness and multiplicity of solutions in standard N -sphere, where N ≥ 5. In the second part of the thesis in Chapter 4, we study problems related to isolated singularities of positive solutions of a polyharmonic operator involving a semilinear term in even dimensions. In particular, we establish: (i) sharp conditions for the extension (and uniqueness) of positive solutions from a punctured domain to its closure, (ii) the regularity of the solutions thus extended.

Acknowledgements First and foremost I offer my sincerest gratitude to my supervisor, Prof. Prashanth K. Srinivasan, who has supported me throughout my thesis with his patience and knowledge whilst allowing me the room to work in my own way, and at the same time the guidance to recover when my steps faltered. Prof. Prashanth taught me how to question thoughts and express ideas. His patience and support helped me overcome many crisis situations and finish this dissertation. One simply could not wish for a better or friendlier supervisor. Besides my advisor, I would like to express my sincere thank to Prof. Mythily Ramaswamy who spend a lot of time in teaching me various courses during my M.Sc and M.Phil. I would like to thank all the teachers here in TIFR-CAM. I have learnt a lot from Prof. Sandeep, Prof. Adimurthi, Prof. Vanninathan, Prof. Datti, Dr. Venky, Dr. Imran, Prof. Mythily, Prof. Vasudeva Murthy and my advisor Prof. Prashanth. They helped me tremendously to develop my background by teaching me so many excellent courses in TIFR-CAM. I take the privilege to thank my under graduation (Bachelor Degree) teachers Dr. Swapan Kumar Chakraborty(SKC), Dr. Subhankar Roy(S.R), Dr. Kartick Chandra Pal(K.P), Lt. Mohanlal Singha Roy(MSR) in Ramakrishna Mission Vidyamandira, Belur Math. I would take this opportunity to thank all my math teachers in Prabeen Sir, Manoj Da, Amiya sir, Lt. Gopi babu for their care in the initial days. I would not have been able to become a part of TIFR-CAM without their guidance. I am privileged to mention the name of Swami Mahamedhananda for providing philosophical and spiritual strength throughout my career. I thank Dr. Dhanya and Dr. Sanjiban Santra for allowing me to write our collaborative work in my PhD thesis. I would also like to thank Dhanya di for the mathematical discussions that we had during my PhD. There are many non-academic staffs of TIFR-CAM who has helped a lot to make my six years of life here so memorable. I thank all the staff members of TIFR headed by Mr. C.J Kannan (and during our initial days by Mr. T.S. Viswanathan) for providing us a very good atmosphere to live and work here. Most of the staffs considered us as their own family members and we could approach them for any ix

help. In this regard I would like to especially mention the names of Desiah, Veena, Joyce and Pramila. Many friends have helped me stay sane through these past few years in the beautiful city Bangalore. A special mention goes to my big group Lali, Debayan, Ali(Kalo), Debabrata(Debu), Indranil(Chaga), Debdip(Guju), Ananta(Prabhu), Rajib da(Katta), Swarnendu(Sil), Arnab da(Dumba da), Mrinmay da(Minu da), Prosenjit da(Lyada), Manas(Manse), Shyam da, Shirshendu da(Kaka da), Bhakti da, Anupam da, Ujjwal Da, Neelabja for making my TIFR life more enjoyable staying far away from home sweet home. Next I would like to thank my fellow batch mates in TIFR-CAM: Saikat(Lali), Debayan, Swarnendu, Debanjana, Mandira, Santanu, Tiju, Raj, Sushobhan for making M.Sc days memorable. Special thanks goes to my college friends Bankim(Banka), Sayan, Birupaksha(Biru), Arindam(BK), Sumit(Bagh), Somnath(Majhi), Sukarna(Captain), Arnab(Mota), Asim(Brad Pitt/Probhu), Soham(Chulki), Pritam(Keira), Suddhasil, Amartya, Bibekanada, Subarna(Ghanta) for their constant encouragements. I’ll definitely cherish all the moments had spent during my college days. I also thank my dear friends Debabrata, Sujoy, Bhaskar, Sayan, Sayantan from my school days, for being such wonderful friends and roomies. In my personal life I have been blessed with a friendly and cheerful family members. First of all my eternal gratitude goes to my Didi(elder sister) Arpita for being a pillar throughout my life. I would like to thank my brother Bunu da(Subhankar) being an inspiration to me in my early days in study. Special thanks to my cousins Lipika(Lipi), Rima, Mridula(Sonu), Biplab(Bipu) and Vikram(Kalu) my uncles and aunts and Kaushik da, Subhra boudi, my nephew Abir. In these acknowledgments, I could not unfortunately cite a large number of people that I met daily or occasionally. Last but not the least I express my deepest thank to my Parents for supporting me throughout my academic career and my personal life. Neither words nor deeds would ever suffice to thank them for their affection for me.

Contents Declaration of Authorship

iii

Abstract

vii

Acknowledgements

viii

1 Introduction 1.1 The Problem of Prescribing the Curvature . . . . . . . . . . . . . 1.1.1 Prescribing Gaussian and Scalar Curvature . . . . . . . . 1.1.2 The Problem of Prescribing the Q-Curvature . . . . . . . 1.1.3 The (PQC) Problem on S4 : Existence and Multiplicity in the Perturbative Case . . . . . . . . . . . . . . . . . . . . 1.1.4 The (PQC) Problem on SN , N ≥ 5: Uniqueness and Exact Multiplicity in the Perturbative Case . . . . . . . . . . . . 1.2 Isolated Singularities for the Biharmonic operator . . . . . . . . . 1.3 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . 2 Perturbed Q-Curvature Problem on the Standard Sphere S4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Solving the reduced Operator Equation . . . . . . . . . . . . . . 2.4 Existence of Solution . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Necessary Condition . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Local Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . 2.7 Exact Multiplicity Result . . . . . . . . . . . . . . . . . . . . . . 2.8 A Concrete Approach for Finding Stable Zeroes of V0 . . . . . .

. . . . . . . .

. . .

1 1 1 3

.

5

. 9 . 13 . 17

. . . . . . . .

19 19 27 32 35 40 45 48 49

3 Exact Multiplicity for the Perturbed Q-Curvature Problem in RN , N ≥ 5 55 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 xi

xii 3.2 3.3 3.4

Preliminary Results. . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Local Uniqueness of Solutions. . . . . . . . . . . . . . . . . . . . . . 63 Exact Multiplicity Result . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Isolated Singularities of Higher Order Elliptic Operators 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Biharmonic Operator in R4 . . . . . . . . . . . . . . . . . . 4.4 Polyharmonic Operator in R2m . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

69 69 71 78 84

. . . .

5 Conclusion 89 5.1 Q-Curvature Problem on the Standard Sphere . . . . . . . . . . . . 89 5.2 Isolated Singularities of Polyharmonic Operator . . . . . . . . . . . 90 6 Appendix 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Lyapunov-Schmidt (or Finite Dimensional) Reduction . . . . . . . . 6.2.1 An Overview of The Reduction Method in Infinite Dimensions 6.3 Kelvin Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography

93 93 93 94 98 99

103

Chapter 1 Introduction In this short introduction, we make an attempt to describe briefly how the various theorems proved in the chapter 2-4 in the thesis. In the last section of the thesis we recall some required basic definitions and important theorems.

1.1

The Problem of Prescribing the Curvature

(background material to Chapters 2 and 3 ) In this section we recall the well known problems of prescribing Gaussian and Scalar curvature on SN for the Laplacian and the so-called Q-curvature for a fourth order conformally invariant elliptic operator. We briefly review major results obtained on these very important problems and state the existence and multiplicity results we have obtained for the prescribed Q-curvature problem.

1.1.1

Prescribing Gaussian and Scalar Curvature

Let (M 2 , g0 ) be a compact closed two-dimensional surface endowed with a smooth metric g0 whose Gaussian curvature is denoted as K0 (which maybe assumed to be a constant by the uniformisation theorem). For a smooth conformal metric g = e2w g0 on M 2 , the Laplace-Beltrami operator transforms according to the rule ∆g = e−2w ∆g0 .

1

(1.1)

1. Introduction

2

We can now state the Prescribed Gaussian Curvature (PGC) problem: Given a smooth function K on (M 2 , g0 ), can we find the conformal factor w so that the resulting metric g has the prescribed Gaussian curvature Kg ≡ K? This is equivalent to solving the following equation for w: − ∆0 w + K0 = Ke2w .

(1.2)

Existence of such a conformal metric g implies the invariance of the total Gauss curvature integral Z

Z Kdµg =

M2

2w

Z

Ke dµg0 = M2

K0 dµg0 M2

under conformal changes of the metric. In fact, by the Gauss-Bonnet formula Z M2

Kdµg = 2πχ(M 2 ) = 4π(1 − ge(M 2 )),

(1.3)

where χ(M 2 ) is the Euler characteristic and ge be the genus of M 2 . Hence the total R Gauss curvature M 2 Kdµg is conformally invariant, and its sign is determined by the sign of χ(M 2 ). The Gauss-Bonnet formula (1.3) gives the first necessary condition for the solvability of (1.2) for a given K. When χ(M 2 ) = 0, the (PGC) problem has been solved completely by KazdanWarner in [42] where it is shown that (1.2) has a solution w if and only if either R (i) K ≡ 0 or (ii) K changes sign with M 2 Ke2f dµg0 < 0, where f is a solution of ∆0 f = K0 .

If χ(M 2 ) > 0 then either M 2 is the real projective plane RP 2 or the standard 2-sphere S2 . The case M 2 = RP 2 has been studied in [5] and [54] where it is shown that K being positive somewhere on M 2 is both necessary and sufficient to solve the (PGC) problem. When M 2 = S2 , the situation is much more complicated and the PGC problem (referred to commonly as Nirenberg problem) has been studied in [21], [22]. In [21],

1. Introduction

3

it is shown that if the given function K is positive in M 2 with only non-degenerate critical points, has at least two local maxima and at the saddle-points of K we have ∆g0 K > 0 then (1.2) has a solution. The sharpness of these assumptions has been shown in [66]. Given a smooth manifold (M N , g0 ) of dimension N ≥ 3, there is an analogous problem called the

Prescribed Scalar Curvature (PSC) Problem: Given a smooth function R on (M N , g0 ), can we find a conformal metric g that has the prescribed scalar curvature Rg ≡ R? 4

Let (SN , g0 ) be the standard N -sphere. If we write g = u N −2 g0 , the (PSC) problem is equivalent to finding a positive function u on SN which satisfies the following equation: N +2

− ∆g0 u + cN R0 u = cN R(x)u N −2 , where cN =

N −2 , R0 4(N −1)

(1.4)

= N (N − 1) is the scalar curvature of (SN , g0 ).

In solving prescribed scalar curvature problem Kazdan and Warner [43] found an obstruction to solving (1.4) in case of sphere SN with standard metric g0 having R0 = N (N − 1), and asserts that any solution of (1.4) satisfies the identity Z

2N

X(R)u N −2 dV0 = 0

(1.5)

SN

where X is any conformal vector field on SN . In particular if we take X to be the gradient of any first-order spherical harmonic ψ and R = ψ + constant, then LHS of (1.5) is positive consequently (1.4) has no solution in this situation. For more details on the (PSC) problem see [44], [45], [24], [19], [25], [76].

1.1.2

The Problem of Prescribing the Q-Curvature

For fourth order elliptic operators, we can study an analogous prescribed curvature problem which we describe below. Let (M 4 , g) be a smooth four dimensional manifold. If we denote by Sg the scalar curvature of g, Ricg the Ricci curvature of g, and ∆g = divg ∇g the Laplace-

Beltrami operator on M 4 , then the Q curvature of (M 4 , g) is defined by the expression Qg = −

1 (∆g Sg − Sg2 + 3|Ricg |2 ). 12

(1.6)

1. Introduction

4

The Q-curvature can in fact be defined on any even dimensional manifold in a similar fashion and it can be shown that the Q-curvature is essentially the Gaussian curvature if M 4 is a surface. Similar to the Gauss or Scalar curvature, the Q-curvature is a conformal invariant and its integral gives information on the topology of the manifold. We recall the definition of the Paneitz operator Pg which was introduced by Paneitz

in [58]:

Pg4 ψ

=

∆2g ψ

+ divg

2 Sg g − 2Ricg dψ, ∀ψ ∈ C ∞ (M 4 ) 3

(1.7)

where d is the de Rham differential of the metric g. Similar to (1.1) the Paneitz operator is conformally invariant on 4-manifolds, in the sense that Pgw (ψ) = e−4w Pg (ψ)

for all

ψ ∈ C ∞ (M 4 ),

for any conformal metric gw = e2w g. For more details about the basic properties of the Paneitz operator one can see [17], [23], [13], [18], [20], [39]. Analogous to the identity (1.2), it was shown by Branson-Orsted [14] that the Qcurvature of a metric g = e2w g0 is related to the Q-curvature Q0 of the background metric g0 via the equation Pg0 w + 2Q0 = 2Qe4w .

(1.8)

Finally if we denote by W the Weyl tensor of M , then similar to the Gauss-Bonnet formula (1.3) there holds the identity (see [10]) |W |2 Q+ dµg = 4π 2 χ(M 4 ). 8 4 M

Z

(1.9)

If M 4 is a locally conformally flat manifold, then W ≡ 0, and we obtain the exact

analogue of the Gauss-Bonnet formula.

The Paneitz operator was generalized to higher dimensions by Branson [12]. Given a smooth compact Riemannian N -manifold (M N , g), N ≥ 5, the Branson-Paneitz operator is defined as

PgN ψ = ∆2g ψ − divg (aN Sg g + bN Ricg )dψ +

N −4 N Qg ψ, 2

(1.10)

1. Introduction

5

where (N − 2)2 + 4 4 , bN = − , 2(N − 1)(n − 2) N −2 1 N 3 − 4N 2 + 16N − 16 2 2 N Qg = − ∆g Sg + Sg − |Ricg |2 . 2 2 2 2(N − 1) 8(N − 1) (N − 2) (N − 2) aN =

4

See [28] for more details about the properties of PgN . If gv = v N −4 g is a conformal

metric to g, one has that for all ψ ∈ C ∞ (M ),

N +4

PgN (ψv) = v N −4 PgNv ψ and PgN v =

+4 n − 4 N NN −4 Qgv v . 2

(1.11)

Similar to the (PGC) or the (PSC) problem, in the context of equation (1.8) or (1.11), it is natural to pose the Prescribed Q-Curvature (PQC) problem: Given a smooth function Q on (M N , g0 ), can we find the conformal factor w (or v) so that the resulting metric g has the prescribed Q-curvature Qg ≡ Q? It is obvious that the (P QC) problem is equivalent to asking whether we can solve the fourth order PDEs in (1.8) or (1.11) with the choice Q = Qgw or Q = QN gv respectively. We would like to note that the (PGC), (PSC) problems have been widely studied in [6], [40], [44] and [45]. On the other hand the (PQC) problem has been studied in [7], [29] and [55] on four dimensional Riemannian manifolds and in [8] the (PQC) problem has been treated on Riemannian manifolds of even dimension. We also cite [51] where the (PQC) problem discussed on the standard sphere S4 , [32] which involves the study on SN , N ≥ 5 and [74] which treats the problem on SN , N ≥ 3.

1.1.3

The (PQC) Problem on S4 : Existence and Multiplicity in the Perturbative Case

(A brief discussion of the results contained in Chapter 2)

1. Introduction

6

In the second chapter of the thesis we discuss in detail the existence and exact multiplicity results we have obtained to the (PQC) problem on the round sphere (S4 , g0 ) when the prescribed function Q is a perturbation of a constant. It can be easily checked that Qg0 ≡ 3 and the Paneitz operator takes the simpler form Pg0 = ∆2g0 − 2∆g0 . Given the prescribed smooth function Q on S4 , we are therefore required to solve the fourth order PDE given in (1.8):

(∆2g0 − 2∆g0 )w + 6 = 2Qe4w

on S4 .

(1.12)

Recall the inverse of the stereographic projection Π−1 : R4 → S4 given by x 7→

2x |x|2 − 1 , 1 + |x|2 |x|2 + 1

.

By a direct computation, we see that, Pg0 Φ(u) =

(1 + |x|2 )4 2 ∆ u for all u ∈ C ∞ (R4 ) 16

where Φ(u)(y) = u(x) + log(1 + |x|2 ) − log 2,

Π(y) = x.

Then the equation (1.12) transforms under the stereographic projection Π into 4u ˆ ˆ = Q ◦ Π−1 . ∆2 u = 2Q(x)e in R4 , where Q

(1.13)

We would like to study the (PQC) problem only when Q is a perturbation of a constant function. More precisely, we let Q = 3(1 + h),

h is a smooth function on S4 and > 0 is small.

That is, we transform (1.12) to the following problem ∆2 u = 6(1 + f (x))e4u in R4 ,

(1.14)

1. Introduction

7

where f := h ◦ Π−1 . Note that the problem (1.14) is a perturbation of the following problem 2

∆ U = 6e

4U

Z

4

in R , R4

e4U < ∞

(1.15)

whose solutions are classified by Lin [46] as: Uδ,y (x) = log

δ2

2δ , + |x − y|2

with (δ, y) ∈ R+ × R4 .

(1.16)

We would like to note that it is impossible to solve the problem (1.14) by the standard variational methods staying in R4 as it can be checked that ∆U 6∈ L2 (R4 ).

But using the tool of Lyapunov-Schmidt reduction, we prove the existence of a solution to the above problem. We recall

Definition 1.1.1. (Stable zero of a vector field) A zero (δ, y) ∈ R+ × RN of a vector field ν0 ∈ C(Ω; RN ) is called stable if for any sequence of vector fields

ν ∈ C(Ω; RN ) converging to ν0 in a neighbourhood of (δ, y), there exist a zero (δ , y ) of ν with (δ , y ) → (δ, y) as → 0.

Next we introduce the following weighted Sobolev space with the weight function w(x) = (1 + |x|2 ). Let 4,2 H = u ∈ Wloc (R4 ) : w2 ∆2 u, w|∇(∆u)|, ∆u, w−1 |∇u|, w−2 u ∈ L2 (R4 ) with the inner product hu, viH =

Z

4

2

2

Z

w ∇(∆u) · ∇(∆v) + Z −2 w ∇u · ∇v + w−4 uv.

w ∆ u∆ v + R4Z

2

R4

+ R4

Z ∆u∆v R4

R4

Finally let wδ,y (x) = (δ 2 + |x − y|2 ). We define Hδ,y by replacing the weight w by wδ,y in the definition of H.

We denote the derivatives of Uδ,y as follows (i = 1, 2, 3, 4): (0)

ψδ,y (x) =

∂Uδ,y , ∂δ

(i)

ψδ,y (x) =

∂Uδ,y . ∂xi

1. Introduction

8

The solutions of (1.15) form a five dimensional manifold which we denote by M = {Uδ,y : (δ, y) ∈ R+ × R4 }. For any compact set K ⊂ R+ × R4 define d(u, MK ) = inf ku − Uδ,y kH1,0 , (δ,y)∈K

where MK = {Uδ,y : (δ, y) ∈ K}.

Let the vector field ν0 : R+ × R4 → R5 be defined as Z ν0 (δ, y) =

4Uδ,y

f (x)e R4

(0) ψδ,y (x)dx, ...,

Z f (x)e

4Uδ,y

R4

(4) ψδ,y (x)dx

.

We note that ν0 (δ, y) = ∇J(δ, y) where J(δ, y) =

Z

f (x)e4Uδ,y dx,

R4

and hence ν0 is a gradient vector field. We show that a stable zero (δ, y) ∈ R+ ×RN

of ν0 will make the corresponding Uδ,y a bifurcation point for a continuum of solutions to (1.14) as → 0. The main results we obtain are Theorem 1.1.1. (ref. [59])(“Bifurcation”from a stable zero) Let K ⊂ R+ × R4

be a compact set with a nonempty interior. Let (δ, y) ∈ K be a stable zero of the

vector field ν0 . Then there exists a 0 > 0 depending on K such that (1.14) admits a solution u for all ∈ (0, 0 ). Moreover, u = Uδ ,y + φ with kφ kHδ,y = O() and (δ , y ) → (δ, y) as → 0.

Remark 1.1.1. We note that the above result is based only on the shape of the function f near the critical points (for more concrete examples see Section 2.8) whereas the previous works are based on the non-degeneracy of the critical points.

Theorem 1.1.2. (ref. [59])(Necessary condition) Let u be a sequence of solution of (1.14) such that ku − Uδ,y kHδ,y → 0. Then ν0 (δ, y) = 0. Theorem 1.1.3. (ref. [59])(Local uniqueness) Let K ⊂ R+ × R4 with a nonempty

interior. Let (δ, y) ∈ K be a zero of the vector field ν0 such that D2 J(δ, y) is invertible. Furthermore, suppose f satisfies

|∇f (x)| ≤ C.

1. Introduction

9

If {u,i }, i = 1, 2 are two sequences of solutions of (1.14) such that ku − Uδ,y kHδ,y → 0 as → 0, then there exists 0 (K) > 0 depending on K such that for all ∈ (0, 0 ) we obtain u,1 ≡ u,2 .

Theorem 1.1.4. (ref. [59])(Exact multiplicity) Let ν0 have only finitely many zeros all of are stable and contained in a compact set K ⊂ R+ × R4 . Suppose

that at any stable zero of ν0 the Hessian D2 J is invertible. Then there exists a

ρ0 = ρ0 (K) > 0 and 0 = 0 (ρ0 ) > 0 such that for all ∈ (0, 0 ), the problem (1.14)

has exactly the same number of solutions u with d(u, MK ) < ρ0 as the number of stable zeros of ν0 .

The proof of the existence theorem is achieved, as remarked earlier, through the method of finite dimensional reduction and the uniqueness theorem uses in an essential way the Pohazaev type identities associated to the equation in (1.14). We also remark that for showing existence, all the earlier works (ref. [8], [18], [51], [66], [74], [75]) assume some kind of non-degeneracy condition on the function Q. Our assumption that the zeroes of the vector field ν0 are stable is more general and implies some restriction only on the shape of Q near its critical points.

1.1.4

The (PQC) Problem on SN , N ≥ 5: Uniqueness and

Exact Multiplicity in the Perturbative Case (A brief discussion of the results contained in Chapter 3)

In the third chapter of the thesis we prove an exact multiplicity result for the conformally invariant fourth order equation (1.11) on the round sphere (SN , g0 ), N ≥

5, in the perturbative case:

Q = 1 + K. Recall the definition of Branson-Paneitz operator from (1.10). It is easy to check that on the unit sphere (SN , g0 ) with g0 the standard metric, PgN0 v = ∆2g0 v − cN ∆g0 v + dN v

(1.17)

1. Introduction

10

where 1 cN := (N 2 − 2N − 4); 2 1 dN := N (N − 4)(N 2 − 4). 16 Define D2,2 (RN ) to be the completion of C0∞ (RN ) in the norm kukD2,2 (RN ) :=

Z

2

RN

21

|∆u|

.

It is worth noting that the PDE in RN obtained via the stereographic projection of (1.17) possesses an energy functional in the Sobolev space D2,2 (RN ) and hence

falls into the framework of standard Variational methods. An existence result for this problem has been shown in V. Felli [32] where she adapts the arguments of [3] to the Branson-Paneitz operator. In particular, [32] uses the perturbed critical point theory introduced in [1] and [2]. Let us denote by δ the Euclidean metric on RN and by Π the stereographic projection from SN to RN . Then we have 4

(Π−1 )∗ g0 = ϕ N −4 δ where ϕ(y) =

2 1 + |y|2

N2−4 .

Now we define Φ : D2,2 (RN ) → H 2 (SN ) as Φ(u)(x) =

u(Π(x)) . ϕ(Π(x))

It can be shown that Φ is actually an isomorphism between H 2 (SN ) and D2,2 (RN ).

Since the Branson-Paneitz operator is invariant under a conformal change of metric, we have N +4

N +4

PgN0 (Φ(u)) = ϕ− N −4 PδN (u) = ϕ− N −4 ∆2 u ∀u ∈ C ∞ (RN ). By the transformations ˆ := Q ◦ Π−1 , v := Φ(u) and Q

(1.18)

1. Introduction

11

the equations (1.11)-(1.17) get simplified into the following equation (

∆2 u = u > 0,

N +4 N −4 ˆ N Qu −4 2 N

(1.19)

in R .

We now assume that Q is a perturbation of the constant, which leads to the following form ˆ = 1 + K, Q

K ∈ L∞ (RN ) ∩ C 2 (RN ).

Then the equation (1.19) leads us to consider the following problem with K ∈ C 2 (RN ), for > 0:

(P )

    

Find u ∈ D2,2 (RN ) solving:  N +4 2  N −4 ∆ u = (1 + K(x))u   in RN .    u>0

Using techniques introduced in [1] and [2], existence of solutions to the problem (P ) was proved in [32] with the following assumptions on K: (K1) K ∈ C 2 (RN ), kKkL∞ (RN ) + k∇KkL∞ (RN ;RN ) + kD2 KkL∞ (RN ;RN ×RN ) < ∞. (K2) (a) There exists ρ > 0 such that h∇K(x), xi < 0, R (b) h∇K(x), xi ∈ L1 (RN ), RN h∇K(x), xidx < 0.

|x| ≥ ρ,

(K3) The set of all critical points of K, denoted by crit(K), is finite. (K4) ξ ∈ crit(K), there exists β = βξ ∈ (1, N ) and aj ∈ C(RN ), 1 ≤ j ≤ N , such P P that Aξ := j aj (ξ) 6= 0. Furthermore, K(y) = K(η) + j aj |y − η|β + o(|x − y|β ) as y → η for any η in a small neighbourhood of ξ.

(K5)

P

Aξ 0 such that the following hold: (i) (P ) has a solution ∀ ∈ (0, 0 ), (ii) (see [2]) ∀ ∈ (0, 0 ), there exists (µ , ξ ) ∈ R+ × RN such that for any compact set A ⊂ R+ × RN , we can find a constant c(A) > 0 such that ku − zµ ,ξ kD2,2 (RN ) ≤ c(A). (iii) (µ , ξ ) → (µ, ξ). Define

d(u, MA ) = inf ku − zµ,ξ kD2,2 (RN ) for u ∈ D2,2 (RN ). (µ,ξ)∈A

Now we state the following local uniqueness and exact multiplicity results proved in [62] Theorem 1.1.5. (Local Uniqueness) Let (µ, ξ) ∈ R+ × RN be a stable zero of ∇Γ

in the sense that D2 Γ(µ, ξ) is invertible . Let {u1, }>0 and {u2, } be two sequences

of solutions to (P ) with kui, − zµ,ξ kD2,2 (RN ) → 0 as → 0, i = 1, 2. Then there exists 0 such that u1, = u2, , ∀ ∈ (0, 0 ).

1. Introduction

13

The proof of the above local uniqueness theorem is done, as in the case of four dimensions, using Pohazaev type identities that hold for the PDE in (P ). Theorem 1.1.6. (ref. [62])(Exact Multiplicity) Let K satisfies the assumptions (K1)-(K5). We further suppose that V0 has finitely many zeroes in R+ × RN all of

which are stable in the sense that the derivative Dµ,ξ V0 is invertible at each zero . Let A ⊂ R+ ×RN be any compact set containing the zeroes of V0 . Then there exists

ρ0 = ρ0 (A) > 0 and 0 = 0 (ρ0 ) > 0 such that for all ∈ (0, 0 ), the equation (P ) has exactly the same number of solutions u with d(u, MA ) < ρ0 as the number of zeroes of V0 .

The above theorem is obtained as a consequence of the local uniqueness result in Theorem 1.1.5.

1.2

Isolated Singularities for the Biharmonic operator

(A brief discussion of the results contained in Chapter 4 ) Let Ω ⊂ RN a bounded domain containing the origin. We denote the punctured

domain by Ω0 := Ω \ {0}. Let

f : IR → [0, ∞) be a continuous map. We are interested in the study of the equation ∆2 u = f (u) in D0 (Ω0 ). The main questions we ask are: Given ∆2 u = f (u) in D0 (Ω0 ) with u ≥ 0 and −∆u ≥ 0, what is the equation satisfied by u in the entire domain Ω and how regular is u there?

This type of problem has been first studied by Brezis and Lions [15] in the context of Laplace operator and they proved the following result :

1. Introduction

14

Theorem 1.2.1. Assume that f satisfies lim inf t→∞

f (t) > 0. t

Let u ∈ L1loc (Ω0 ), u ≥ 0, be such that f (u) ∈ L1loc (Ω0 ) and solves −∆u = f (u) in

D0 (Ω0 ). Then, f (u) ∈ L1loc (Ω) and for some α ≥ 0, u solves −∆u = f (u) + αδ0 in D0 (Ω).

As an application of this result P.L. Lions [49] studied the model problem −∆u = up in Ω 1 ≤ p < ∞. For N ≥ 3, he found that p = p >

N , N −2

N N −2

is the critical exponent in the sense that if

then α = 0 in theorem 1.2.1 and if p
0. N

For nonlinearity f (u) growing like the critical power u N −2 , these two phenomena coexist (see [57]). In [26], the authors further extended the above results for dimension N = 2 by finding the optimal growth rate of the function f which ensures that α = 0 or otherwise. We next come to the case of biharmonic operators. We assume (H1) f : [0, ∞) → [0, ∞) is a continuous nondecreasing function with f (0) = 0, (H2) a(x) is a non-negative measurable function belonging to Lk (Ω) for some k > 34 , (H3) there exists r0 > 0 such that Br0 b Ω and essinfBr0 a(x) > 0. Let u be a function which solves the following problem: ( (P )

∆2 u = a(x)f (u) in D0 (Ω0 ),

u≥0 ,

−∆u ≥ 0 in Ω0 .

We have the following analogue of theorem 1.2.1:

1. Introduction

15

Theorem 1.2.2. Suppose u, ∆u and ∆2 u ∈ L1loc (Ω0 ). Let ∆2 u = a(x)f (u) in D0 (Ω0 ) with u ≥ 0 and −∆u ≥ 0 a.e in Ω0 . Then u, a(x)f (u) ∈ L1loc (Ω) and there

exist non-negative constants α, β such that ∆2 u = a(x)f (u)+αδ0 −β∆δ0 in D0 (Ω). That is, u is a distributional solution of the problem:  2    ∆u

= a(x)f (u) + αδ0 − β∆δ0

u ≥ 0, −∆u ≥ 0

(Pα,β )

) in Ω,

   α, β ≥ 0; u and a(x)f (u) ∈ L1 (Ω). For the biharmonic operator, we recall the result of Soranzo [65] where the author considers a distributional solution u to the following weighted nonlinear problem in dimensions N ≥ 4: ( (P )

∆2 u = |x|σ up in D0 (Ω0 ), σ ∈ (−4, 0]

u≥0 ,

−∆u ≥ 0 in Ω0 .

He obtains the following result showing when the singularity at 0 is removable for the choice of the exponents: 1 < p
1 and −4 < σ ≤ 0. Then (u, v) ∈ Lloc (Ω) ∩ Mloc (Ω) and |x|σ up ∈ L1loc (Ω). Moreover, there exists two parameters α, β ≥ 0 such that (

−∆u = v + βδ0

in D0 (Ω)

−∆v = |x|σ up + αδ0

in D0 (Ω).

(1.21)

In particular u is a solution of ∆2 u = |x|σ up + αδ0 − β∆δ0

in D0 (Ω).

(1.22)

1. Introduction

16 N

N

N −4 N −2 If in addition β = 0, then u ∈ Mloc (Ω) and −∆u ∈ Mloc (Ω), when N > 4 or

2 u ∈ Lqloc (Ω) for any q < ∞ and ∆u ∈ Mloc (Ω), when N = 4.

Corollary 1.2.1. (ref. [65]) Let N ≥ 4 and let u ∈ C 4 (Ω0 ) satisfies (P ). Then +σ (i) if p ≥ max 1, N ⇒ β = 0; N −2 (ii) if N > 4, p ≥

N +σ N −4

⇒ α = 0 and u is a distribution solution of (P ) in Ω.

For the dimension N = 4, we are able to improve (see [27] the above result of Soranzo to a more general exponential type nonlinearity replacing the power-type nonlinearity up . In this work we answer the following questions (see theorem 1.2.4 below): (1) Can we find an optimal growth condition for f near infinity such that β = 0? (2) Can we find a sharp condition on the growth of f at infinity that determines whether both α and β are zero, and in such a case is it true that u is regular in Ω? We define the following growth condition: Definition 1.2.1. We call f a sub-exponential type function if lim sup f (t)e−γt ≤ C for some γ, C > 0. t→∞

We call f to be super-exponential type if it is not a sub-exponential type function. We can show that if f (t) grows at least like t2 near infinity, then β = 0, and if f is a super-exponential type function, then α, β = 0. Theorem 1.2.4. (see [27]) 1. (Removable Singularity) Let f be a super-exponential type function also a and f satisfy hypotheses (H1)-(H3). Then any solution u of (P ) extends to a distributional solution of (P0,0 ). 2. (Existence of singular solutions) Let f be a sub-exponential type nonlinearity, also f and a satisfy the hypotheses (H1)-(H3). Additionally assume that lim inf t→∞

f (t) = c ∈ (0, ∞]. t2

1. Introduction

17

Then there exists an α∗ = α∗ (γ) > 0, (γ appearing from the fact that f is subexponential) such that for all 0 < α ≤ α∗ the problem (Pα,0 ) admits a solution in Br (0).

3. (Regularity in Ω) Let f be a sub-exponential type function satisfying (H1)(H3). Let u be a solution of (P0,0 ) with u = ∆u = 0 on ∂Ω. Then, u is regular in Ω. There are examples of unbounded solutions to (P0,0 ) in case f is of superexponential type. The proof of the removable singularity is similar to the one in higher dimensions by Soranzo [65]. The existence of singular solution to (Pα,0 ) is proved using the method of sub and super solutions, where 0 is the sub solution and a super solution is constructed by a perturbation of the fundamental solution. The regularity of distributional solution to (P0,0 ) is different since the usual bootstrap argument does not work. Instead, we use some results in Brezis and Merle [16] to obtain the regularity of u. In [27] we also extend the above results to the polyharmonic operator of order 2m in R2m .

1.3

Mathematical Preliminaries

In this section we recall some basic definitions and theorems required in the thesis. Theorem 1.3.1. (Implicit Function Theorem) Let E1 , E2 and F be normed linear spaces and assume that E2 is complete. Let Ω ⊂ E1 × E2 be an open set and f : Ω → F be a function such that (1) f is continuous; ∂f (2) for every (x1 , x2 ) ∈ Ω, ∂x (x1 , x2 ) exists and is continuous on Ω; 2

(3) f (a, b) = 0 and A =

∂f (a, b) ∂x2

is invertible with continuous inverse.

Then, there exist neighborhood U of a and V of b and a continuous function ϕ : U → V such that ϕ(a) = b and f (x, ϕ(x)) = 0 and these are the only solutions of the equation f (x, y) = 0 in U × V.

1. Introduction

18

Definition 1.3.1. (Contraction Mapping) Let (X, d) be a metric space. A map f : X → X is called a contraction mapping if there exists a constant 0 < λ < 1 such that

d(f (x), f (y)) ≤ λd(x, y)

∀x, y ∈ X.

Theorem 1.3.2. (Banach Fixed Point Theorem) Let f be a contraction map on a complete metric space X. Then f has a unique fixed point theorem x¯ ∈ X. Definition 1.4. (Fredholm Operator) Let X and Y be Banach spaces on R and T be a bounded linear operator in L(X, Y ). Then T is a Fredholm operator of index p if : (1) dimN (T ) < ∞. (2) R(T ) is closed and codimR(T ) < ∞. (3) p = dimN (T ) − codimR(T ). Where N (T ) and R(T ) denote kernel and range of T respectively. Definition 1.5. (Bifurcation Point)Suppose X, Y be Banach spaces on R and F ∈ C 1 (R × X, Y ). We also assume F (λ, 0) = 0 for all λ ∈ R. Then, (λ0 , 0) is a

bifurcation point if there exists a sequence {(λn , xn )}n such that 1. F (λn , xn ) = 0 and xn 6= 0 for all n. 2. λn → λ0 and x0 → 0.

Definition 1.5.1. (Conformally Invariant Operator)On a general Riemannian manifold M with metric g, a metrically defined operator A is said to be conformally invariant if, under the conformal change in metric gw = e2w g, the pair of corresponding operators Aw and A are related by Aw (φ) = e−bw A(eaw φ) for all φ ∈ C ∞ (M ) and some constants a and b.

Chapter 2 Perturbed Q-Curvature Problem on the Standard Sphere S4 2.1

Introduction

In this chapter we study existence and local multiplicity of solutions to a fourth order equation arises from conformal geometry. We also establish a local uniqueness result. Fourth order operators arise in the applications in the areas of conformal geometry, thermionic emission, gas combustion and gauge theory, plate equation. In the search for a higher order conformally invariant operator Paneitz discovered an interesting fourth order conformally covariant operator on a compact 4-manifold. Let (M, g) be a Riemannian manifold with dim(M ) = 4. Let ∆g be the Laplace Beltrami operator, divg the divergence operator, d the differential and Sg , Ricg denote the scalar curvature and Ricci tensor of the metric g respectively, then the Paneitz operator Pg can be written in the form Pg ψ =

∆2g ψ

+ divg

2 Sg g − 2Ricg dψ, 3

where ψ ∈ C ∞ (M ) (see Paneitz [58], Chang- Yang [23]). The regularized determinant of the Paneitz operator arises in quantum gravity. The Paneitz operator has been most thoroughly studied in dimension four where

19

2. Q-Curvature Problem on S4

20

it appears naturally in connection with extremal problems for the functional determinant of the Laplacian. If dim(M ) = 4, the analogue of the Gauss curvature for a surface is the so-called Q-curvature function given as:

Qg = −

1 (∆g Sg − Sg2 + 3|Ricg |2 ). 12

In fact, Paneitz operator was generalized by T. Branson for N ≥ 3 (see [12]).

Let us now consider the question:

Given a smooth function Q on S4 , does there exist a metric g conformal to the standard metric g0 such that Q = Qg ? If we assume a conformal transformation of the form g = e2w g0 , the answer to the above question is “yes” iff we can solve for w in the equation Pg0 w + 2Qg0 = 2Qe4w on S4 . It can be checked that Qg0 ≡ 3 and that the Paneitz operator on (S4 , g0 ) is given by Pg0 = ∆2g0 − 2∆g0 . Hence, we look to solve for w in the problem: ∆2g0 − 2∆g0 w + 6 = 2Qe4w on S4 .

(2.1)

Integrating (2.1) over S4 , one obtains that the total Q− curvature of (S4 , g0 ) denoted by kg0 , which is a conformal invariant, satisfies Z kg0 =

Qe S4

4w

Z =

Qg0 = 3vol(S4 ).

S4

Furthermore, if g is conformal to g0 , the Weyl tensor of (S4 , g) vanishes identically and the following Gauss-Bonnet type formula holds: Z

Qg = 4π 2 χ(S4 ) = 8π 2

(2.2)

S4

where χ is the Euler characteristic. This immediately gives the first obstruction: If Q ≤ 0, then (2.1) has no solution. More subtle obstructions similar to the Kazdan-Warner identities can be shown in the case of (2.1) as well (see section 2.5

for details). The problem (2.1) is variational and the solutions can be characterized


21

as critical points of the following functional on H 2 (S4 ): 1 J(u) = vol(S4 )

Z S4

(uPg0 u + 4u)dµg0 − 3 log

1 vol(S4 )

Z

4u

Qe dµg0 . S4

However, the functional fails to satisfy Palais-Smale condition. Hence, for these reasons, solvability of (2.1) is not straight forward. Using ideas similar to the ones used in [21], [22] and [18] to solve the Nirenberg’s problem on SN , Wei-Xu [74] proved existence of solutions of (2.1) when Q > 0 satisfies the nondegeneracy condition (∆Q(x))2 + |∇Q(x)|2 6= 0,

(2.3)

and the vector-field G : SN → RN +1 defined by G(x) = (−∆Q(x), ∇Q(x)).

(2.4)

This mapping is well defined and it never vanishes if the condition (2.3) is verified. Thus

G |G|

from S4 to S4 will be well defined. If Q is C 3 function, then

G |G|

is C 1 on

G S4 , hence its degree deg( |G| is well defined. In [74] it was shown that if Q > 0 on G S4 is non-degenerate and deg( |G| , SN ) 6= 0, then (2.1) has a solution. Later, in the

work [74], they extended their results to very general pseudo-differential operators N

on SN which look like (−∆) 2 when N is odd. To our knowledge it seems that the non-degeneracy condition (2.3) is crucially required in [18], [73] and [74] to obtain apriori estimates for the solution of (2.1). The other approach is via the heat-flow as done in [66], [8] and [51]. In particular, Malchiodi-Struwe ([51]), proved existence of a solution to (2.1) assuming that Q is a Morse function (i.e., has only nondegenerate critical points p) with Morse Index ind(Q, p) such that ∆Q(p) 6= 0 and satisfies the index count X

(−1)ind(Q,p) 6= 1.

∇Q(p)=0,∆Q(p) 0 is a small parameter. Using stereographic projection from S4 to R4 , we transform (2.1) (with f denoting the transformed function h) to the following problem:

∆2 u = 6(1 + f (x))e4u in R4 .

(2.6)

Note that the problem (2.6) is a perturbation of the following problem   

∆2 U = 6e4U in R4 , Z

 

e4U < +∞

(2.7)

R4

whose solutions in the space E (see below for definition of E) are classified by Lin [46] as : Uδ,y (x) = log

δ2

2δ , + |x − y|2

with (δ, y) ∈ R+ × R4 .

(2.8)

We remark that, if U = U1,0 solves (2.7), then so does the function w(x) = U1,0 ( |x|x2 ) − 2 log |x|.


23

In this work, taking advantage of the fact that we are in a perturbative situation, we show existence of a solution to (2.6) without assuming that Q (and hence f ) satisfies the non-degeneracy conditions as in (2.3). In particular, we do not assume Q to be a Morse function. What we assume is something about the “shape” of Q near the critical points (see the definition of the quantity Cβ,ξ in Section 2.8). As in the previous works, the main idea is to define a suitable vector-field V0 on R+ × RN (see (2.10)). A stable zero (see definition 2.1.5) (δ, y) ∈ R+ × RN of V0

will make the corresponding Uδ,y a “bifurcation point” for a continuum of solutions to (2.6) as ε → 0. For a precise statement of this fact see Theorem 2.1.1 below. If we assume that this zero is “stable” in the more standard sense, we can show that this “bifurcation” branch from Uδ,y is locally unique; this also leads to an exact

multiplicity result for (2.6) for all small ε > 0. For a precise statement of such uniqueness and multiplicity see Theorems 2.1.3 and 2.1.4 below. It is not possible to study (2.6) directly in a variational framework since we can check that ∆U 6∈ L2 (R4 ). Due to this fact we will work in a non-variational framework using weighted Sobolev spaces as in [53], [37], [74] to perform the Lyapunov-Schmidt reduction (for details see Chapter 6). Let ω(x) = (1 + |x|2 ). We introduce the following weighted Sobolev spaces: 4,2 4 2 2 −2 2 4 Definition 2.1.1. Let E = u ∈ Wloc (R )|ω ∆ u, ω u ∈ L (R ) equipped with R R the inner product hu, viE = R4 ω 4 ∆2 u∆2 v + R4 ω −4 uv. Definition 2.1.2. Let 4,2 4 2 2 −1 −2 2 4 H = u ∈ Wloc (R )|ω ∆ u, ω|∇(∆u)|, ∆u, ω |∇u|, ω u ∈ L (R ) with the inner product hu, viH =

Z

4

2

2

Z

ω ∆ u∆ v + 4

ZR

+ R4

4

ω −2 ∇u · ∇v +

Definition 2.1.3. ˜ = H

u∈

R Z

ω ∇(∆u) · ∇(∆v) + ω −4 uv.

R4

L2loc (R4 )|ω 2 u

with the inner product hu, viH˜ = Finally,

2

Z R4

2

4

∈ L (R )

ω 4 uvdx.

Z ∆u∆v R4


24

˜ δ,y by Definition 2.1.4. Let ωδ,y (x) = (δ 2 + |x − y|2 ). We define Eδ,y , Hδ,y and H ˜ respectively. replacing the weight ω by ωδ,y in the definitions of E, H and H Remark 2.1.1. It is easy to see that Uδ,y ∈ Eδ,y for all (δ, y). ˜ δ,y are uniRemark 2.1.2. We can easily check that the spaces Hδ,y , Eδ,y and H ˜ respectively as (δ, y) varies over formly equivalent as Hilbert spaces to H, E and H a compact set K ⊂ R+ × R4 . Remark 2.1.3. It is easy to see that Hδ,y is continuously embedded in Eδ,y . We denote the derivatives of Uδ,y as follows (i = 1, 2, 3, 4):  ∂Uδ,y (|x − y|2 − δ 2 ) (0)   ψ (x) = =  δ,y ∂δ δ(δ 2 + |x − y|2 )  ∂Uδ,y 2(xi − yi ) (i)   ψδ,y (x) = =− 2 . ∂xi (δ + |x − y|2 )

, (2.9)

As noted before, the solutions of (2.7) form a five dimensional manifold which we denote by M = {Uδ,y : (δ, y) ∈ R+ × R4 }. For any compact K ⊂ R+ × R4 define d(u, MK ) = inf ku − Uδ,y kH1,0 . (δ,y)∈K

Let the vector-field V0 : R+ × R4 → R5 be defined as V0 (δ, y) =

Z

4Uδ,y

f (x)e R4

(0) ψδ,y (x)dx, · · ·

Z ,

f (x)e

4Uδ,y

R4

(4) ψδ,y (x)dx

.

(2.10)

We note that V0 is a gradient vector field as V0 (δ, y) = ∇J(δ, y) where J(δ, y) =

Z

f (x)e4Uδ,y dx.

(2.11)

R4

We make the following definition of a stable vector-field: Definition 2.1.5. Let Ω ⊂ RN be an open set. We call a point P ∈ Ω as a stable

zero for a vector field V0 ∈ C(Ω; RN ) if V0 (P ) = 0 and for any sequence of vector fields Vε ∈ C(Ω; RN ) converging uniformly to V0 in a neighborhood of P , there exists a zero Pε of Vε with Pε → P as ε → 0.


25

We now state the theorems we will prove. Theorem 2.1.1. (“Bifurcation” from a stable zero) Let K ⊂ R+ ×R4 be a compact

set with a nonempty interior. Let (δ, y) ∈ K be a stable zero of the vector field V0 . Then there exists a 0 > 0 depending on K such that (2.6) admits a solution u for

all ∈ (0, 0 ). Moreover, u = Uδ ,y + φ with kφ kHδ,y = O() and (δ , y ) → (δ, y). Theorem 2.1.2. (Necessary Condition) Let u be a sequence of solution of (2.6) such that ku − Uδ,y kHδ,y → 0. Then V0 (δ, y) = 0. Theorem 2.1.3. (Local Uniqueness) Let K ⊂ R+ × R4 with a nonempty interior.

Let (δ, y) ∈ K be a zero of the vector field V0 (δ, y) such that D2 J(δ, y) is invertible. Furthermore, suppose f satisfies

|∇f (x)| ≤ C.

(2.12)

If {u,i }, i = 1, 2 are two sequences of solutions of (2.6) such that ku − Uδ,y kHδ,y → 0 as → 0, then there exists 0 (K) > 0 depending on K such that for all ∈ (0, 0 ) we obtain u,1 ≡ u,2 .

Theorem 2.1.4. (Exact multiplicity) Let V0 have only finitely many zeroes all of

which are stable and contained in a compact set K ⊂ R+ ×R4 . Suppose that at any

stable zero of V0 the Hessian D2 J is invertible. Then there exists a ρ0 = ρ0 (K) > 0

and 0 = 0 (ρ0 ) > 0 such that for all ∈ (0, 0 ),the problem (2.6) has exactly the

same number of solutions u with d(u, MK ) < ρ0 as the number of stable zeroes of V0 .

Remark 2.1.4. The proof of the above theorems are done using LyapunovSchmidt reduction carried out for the nonlinear solution operator (see (2.25)) ˜ δ,y . The calculations for this reduction is given in between the spaces Hδ,y and H sections 2.2 and 2.3. Remark 2.1.5. Consider the problem ∆2 u = 6e4u + Ψ(x, u) in R4

(2.13)


26

where Ψ : R4 × R+ → R is continuous and twice differentiable in the second variable and satisfies

sup |Ψ(x, u)| + |Ψu (x, u)| + |Ψuu (x, u)| ≤ Ce4u

(2.14)

|∇x Ψ(x, u)| ≤ Ce4u .

(2.15)

x∈R4

and

An inspection of the proof of Theorems 2.1.1 - 2.1.4 show that they hold for the problem (2.13) as well if we replace the vector-field V0 by the following : V˜0 (δ, y) =

Z R4

(0) Ψ(x, Uδ,y )ψδ,y (x)dx, · · ·

Z , R4

(4) Ψ(x, Uδ,y )ψδ,y (x)dx

.

(2.16)

Remark 2.1.6. A similar kind of result was obtained by Grossi [36] for single peak solutions of the subcritical singularly perturbed nonlinear Schrödinger equation  2 ∆u − V (x)u+up = 0    u>0

  

in RN , in RN ,

(2.17)

u ∈ H 1 (RN ).

By exploiting the “shape” of the potential V ∈ C 1 (RN ) near its critical points, the author obtained exact multiplicity results for (2.17) whenever > 0 is sufficiently

small. In addition, if P is a nondegenerate critical point of V, the author showed that there is a unique solution concentrating at P . Remark 2.1.7. Moreover, Theorems 2.1.1-2.1.4 hold for the equation (−∆)m u = (2m − 1)!(1 + f (x))e2mu in R2m

(2.18)

where m ∈ N. The construction of solution follows from Wei-Xu [75]. Remark 2.1.8. The following problem was studied by Felli [32]:  2 N +4 ∆ u = (1 + f (x))u N −4 in RN ,    u > 0 in RN ,

  

u ∈ D2,2 (RN ).

(2.19)


27

for N ≥ 5. Existence to the above problem is shown in [32] assuming a suitable “shape” for f near a critical point. In particular, an expansion of the form f (x) = f (η) +

X

aj |y − η|β + o(|y − η|β ) as y → η,

β ∈ (1, N )

is assumed at a critical point η. We remark that the problem (2.19) is variational and can be handled in the Sobolev space D2,2 (RN ).

2.2

Preliminaries

Let log+ |x| = max{0, log |x|}. Lemma 2.2.1. There exists a positive constant C such that sup |v(x)| ≤ CkvkE (|x| + log+ |x| + 1), ∀v ∈ E,

(2.20)

sup |v(x)| ≤ CkvkH (log+ |x| + 1), ∀v ∈ H.

(2.21)

R4

R4

Proof. Note that the fundamental solution of the biharmonic operator in R4 is given by F (x, y) =

1 1 log . 2 8π |x − y|

For v in E with kvkE = 1 we set ∆2 v = g. By definition of the space E, the ˜ Then we can write v = v0 + v1 where ∆2 v0 = 0 and v1 (x) = function g ∈ H. R F (x, y)g(y)dy. We now estimate, R4 Z |v1 (x)| =

F (x, y)g(y)dy R4 Z 1 | log |x − y|||g(y)|dy ≤ 8π 2 R4 Z 12 Z 12 | log |x − y||2 1 2 4 2 (1 + |x| ) |g(y)| dy ≤ 2 4 8π 2 R4 R4 (1 + |x| ) Z 21 1 | log |y||2 ≤ kvkE dy . 2 4 8π 2 R4 (1 + |x − y| )


28

Let | log |y||2 dy 2 4 R4 (1 + |x − y| ) Z Z | log |y||2 | log |y||2 = dy + dy 2 4 2 4 {|y|≤1} (1 + |x − y| ) {|y|≥1} (1 + |x − y| ) = I1 + I2 . Z

I: =

Now we estimate Z I1 = {|y|≤1}

| log |y||2 dy ≤ C (1 + |x − y|2 )4

Z {|y|≤1}

| log |y||2 dy < +∞.

Also for |y| ≥ 2|x|, we have 1 |y − x| ≥ |y| − |x| ≥ |y| 2 and as a result we must have | log |y||2 dy + (1 + |x − y|2 )4

Z I2 =

Z

{|y|≥1}∩{|y|≥2|x|} + 2

{|y|≥1}∩{|y| 0. Putting together the estimates for I1 , I2

and v0 we get (2.20). If v ∈ H with kvkH = 1, we note that the corresponding biharmonic function v0 ∈ H and hence is uniformly bounded in R4 . The estimate for v1 can be obtained as above to get (2.21).

Lemma 2.2.2. (Nondegeneracy) The kernel of the linearised operator ∆2 − 24e4Uδ,y in Eδ,y is five dimensional and is generated by

∂Uδ,y ∂Uδ,y ∂Uδ,y ∂Uδ,y ∂Uδ,y , , , , . ∂δ ∂x1 ∂x2 ∂x3 ∂x4

Proof. Without loss of generality, let δ = 1 and y = 0. Consider the problem ∆2 ψ − 24e4U ψ = 0, ψ ∈ E1,0 .

(2.22)


29

∞ (R4 ). Now from lemma 2.2.1 we know that Then by boot-strap argument ψ ∈ Cloc

ψ has atmost linear growth at infinity. We claim that ψ has to be bounded in

R4 . Let |ψ| ≤ C|x| for |x| 1. Then define the Kelvin transform (for details see Chapter 6) of ψ be

ˆ ψ(x) =ψ

x |x|2

in R4 \ {0}.

(2.23)

ˆ Then ψ(x) ≤ C|x|−1 near the origin and satisfies ∆2 ψˆ −

1 ψˆ = 0 in R4 \ {0}. (1 + |x|2 )4

(2.24)

∞ But ψˆ ∈ L2loc (R4 ) and hence by regularity ψˆ ∈ Cloc (R4 ). Hence ψˆ is bounded near

the origin and hence ψ is bounded at infinity. As a result, we can apply the method of Lin and Wei, Lemma 2.6 in [47], to conclude the non-degeneracy.

We want to find solutions to (2.6) of the form u = Uδ,y + ϕ such that ϕ → 0 as → 0 in Hδ,y . If we plug this ansatz in (2.6) then we have,

∆2 ϕ = 6e4Uδ,y (e4ϕ − 1) + 6f (x)e4(Uδ,y +ϕ ) . This motivates us to introduce the following nonlinear operator Bδ,y from a small ˜ δ,y ball B around the origin in Hδ,y into H ˜ δ,y Bδ,y : B ⊂ Hδ,y 7→ H given by Bδ,y (v) = ∆2 v − 6e4Uδ,y (e4v − 1) − 6f (x)e4(Uδ,y +v) .

(2.25)

Therefore finding solutions u of (2.6), bifurcating from Uδ,y for some (δ, y) ∈

R+ × R4 is equivalent to proving the following lemma.

Lemma 2.2.3. There exists a suitable value (δ, y) ∈ R+ ×R4 for which there exists

0 > 0 such that we can find ϕ ∈ Hδ,y solving Bδ,y (ϕ ) = 0, for all ∈ (0, 0 ). Furthermore, kϕ kHδ,y → 0 as → 0.

We now show some basic properties of Bδ,y . Lemma 2.2.4. Let Bρ (0) ⊂ Hδ,y . Then for ρ > 0 small enough we have ˜ δ,y . Bδ,y (Bρ (0)) ⊂ H


30

Proof. Let kvkHδ,y < ρ. Then using (2.20), we have Z R4

2

2 4 8(Uδ,y +v)

(δ + |x − y| ) e

≤ C1 ≤ C1

Z R4

Z R4

e8v (δ 2 + |x − y|2 )4 +

ec2 kvkHδ,y (1+log |x|) < +∞ (δ 2 + |x − y|2 )4

˜ δ,y . It follows that B δ,y maps provided ρ is sufficiently small. Hence, e4(Uδ,y +v) ∈ H ˜ Bρ (0) into Hδ,y . Theorem 2.2.5. Let Bρ (0) ⊂ Hδ,y , with ρ > 0 small. Then for any ε > 0, ˜ δ,y ). Bδ,y ∈ C 1 (Bρ (0), H Proof. First we prove that ˜ δ,y ). Bδ,y ∈ C 0 (Bρ (0), H ˜ δ,y Let vn → v in Hδ,y where vn , v ∈ Bρ (0). This implies that ∆2 vn → ∆2 v in H

and vn → v in Cloc (R4 ). Hence, again by the estimate (2.20) and dominated convergence theorem we obtain

˜ δ,y . 6(1 + f (x))e4(Uδ,y +vn ) → 6(1 + f (x))e4(Uδ,y +v) in H Now we prove that Bδ,y is continuously differentiable in Bρ (0). We claim that its derivative is given by

  h(B δ,y )0 (v), hi = ∆2 h − 24(1 + f (x))e4(Uδ,y +v) h in R4  h ∈ Hδ,y , v ∈ Bρ (0). .

(2.26)

˜ δ,y be defined by A (h) = ∆2 h − 24(1 + f (x))e4(Uδ,y +v) h. Then Let Av : Hδ,y → H v Av is a continuous linear map for all v ∈ Bρ (0). To see this, let hn → h in Hδ,y . ˜ δ,y as well as hn → h in Cloc (R4 ). As a result we must have Then ∆2 hn → ∆2 h in H 2

2 4

(δ + |x − y| ) (1 + ≤

f (x))2 e8(Uδ,y +v) h2n

Ckhn k2Hδ,y (1 + log+ |x|)2 (δ 2 + |x − y|2 )4

ec1 kvkHδ,y (1+log

e8v h2n ≤C 2 (δ + |x − y|2 )4

+

|x|)

.


31

Hence by the dominated convergence theorem, for ρ > 0 small enough, ˜ δ,y . e4(Uδ,y +v) hn → e4(Uδ,y +v) h in H This shows the continuity of Av . Now we claim that (Bδ,y )0 (v) = Av . We have δ,y B (v + h) − Bδ,y (v) − Av h = 6e4(Uδ,y +v) (1 + f (x))(e4h − 1 − 4h) ≤ Ce4(Uδ,y +v) e4|h| h2

c1 khkHδ,y (1+log+ |x|)

≤ Ce

khk2Hδ,y (1 + log+ |x|)2 (1 + |x|2 )4−c2 kvkHδ,y

.

This implies for kvkHδ,y and khkHδ,y small kBδ,y (v + h) − Bδ,y (v) − Av hkH˜ δ,y ≤ Ckhk2Hδ,y and hence we obtain the required result. Let K = Ker(B0δ,y )0 (0) and R = Im(B0δ,y )0 (0). Then by Lemma 2.2.2 K=

∂Uδ,y ∂Uδ,y ∂Uδ,y ∂Uδ,y ∂Uδ,y , , , , . ∂δ ∂x1 ∂x2 ∂x3 ∂x4

Define ˜ δ,y : hψ, ζi ˜ = 0; ζ ∈ R}. R⊥ = {ψ ∈ H Hδ,y We define for i = 0, 1, 2, · · · 4 (i)

(i)

−4 Φδ,y = ωδ,y ψδ,y . (0)

(1)

(4)

Lemma 2.2.6. R⊥ = span{Φδ,y , Φδ,y , · · · Φδ,y }. Proof. Let ψ ∈ R⊥ . Then by definition we must have hψ, (B0δ,y )0 (0)ζiH˜ δ,y = 0, for all ζ ∈ C0∞ (R4 ). This implies that in the sense of distribution 4 4 ∆2 (ωδ,y ψ) − 24e4Uδ,y ωδ,y ψ = 0.


32

4,2 4 2 ψ) ∈ ∆2 (ωδ,y By elliptic regularity, ψ ∈ Wloc (R4 ) and from the above equation ωδ,y

4 4 L2 (RN ). Hence ωδ,y ψ ∈ Eδ,y . This implies that ωδ,y ψ ∈ K. We note that C0∞ (R4 ) =

Hδ,y . Conversely, if φ ∈ K, we have hφ, ∆2 ψ−e4Uδ,y ψiL2 = 0 for all ψ ∈ C0∞ (R4 ). As −4 4 a result we must have ωδ,y ψ ∈ K. φ ∈ R⊥ for any φ ∈ K. Hence ψ ∈ R⊥ iff ωδ,y

Now we define the quotient spaces ˜ δ,y = H ˜ δ,y /R⊥ . Mδ,y = Hδ,y /K and M ˜ δ,y is an isomorphism onto. Then (B0δ,y )0 (0) : Mδ,y → M Now we are in situation to apply finite dimensional reduction.

2.3

Solving the reduced Operator Equation

Let PK⊥ and PR denote the projections PK⊥ : Hδ,y → Mδ,y , ˜ δ,y → M ˜ δ,y . PR : H For a ball Bρ (0) ⊂ Mδ,y for ρ > 0 small enough, define the reduced solution operator

˜ δ,y as S δ,y (v) = (PR o B δ,y )(v). Sδ,y : Bρ (0) → M ˜ δ,y ) for small ρ > 0 and for any > 0. Then by Theorem 2.2.5, Sδ,y ∈ C 1 (Bρ (0), M For any φ ∈ Bρ (0), we write Bδ,y (φ) = Bδ,y (0) + (Bδ,y )0 (0)φ + Qδ,y (φ),

(2.27)

4Uδ,y 4φ Qδ,y [e − 1 − 4φ]. (φ) = −6(1 + f (x))e

(2.28)

where

Applying the projection PR on either side of (2.27) we obtain Sδ,y (φ) = Sδ,y (0) + PR ((Bδ,y )0 (0)φ) + PR (Qδ,y (φ)) = Sδ,y (0) + (Sδ,y )0 (0)φ + PR (Qδ,y (φ)).

(2.29)


33

Therefore, solving Sδ,y (φ) = 0.

(2.30)

(2.29) reduces to solving Sδ,y (0) + (Sδ,y )0 (0)φ + PR (Qδ,y (φ)) = 0. We note that (S0δ,y )0 (0) is invertible and (Sδ,y )0 (0) → (S0δ,y )0 (0) in the operator

norm as → 0. Therefore, we also obtain the invertibility of (Sδ,y )0 (0) for all small > 0. Hence, solving (2.30) for small > 0 is equivalent to solving φ = −((Sδ,y )0 (0))−1 [Sδ,y (0) + PR (Qδ,y (φ))].

(2.31)

Motivated by the above equation, define the map Gδ,y : Bρ (0) → Mδ,y by Gδ,y (v) = −((Sδ,y )0 (0))−1 [Sδ,y (0) + PR (Qδ,y (v))].

(2.32)

Then solving (2.30) for small > 0 is equivalent to finding a fixed point of the map Gδ,y . We do so in the lemma below, thereby solving the reduced operator equation: Lemma 2.3.1. Let K be a compact subset of R+ × R4 and ρ > 0 be small. Then

there exists 0 = 0 (K, ρ) > 0 such that for all ∈ (0, 0 ) and (δ, y) ∈ K, there

∈ Bρ (0) of the map Gδ,y . That is, Sδ,y (φδ,y exists a fixed point φδ,y ) = 0 for all ∈ (0, 0 ), (δ, y) ∈ K.

Proof. We use Banach fixed point theorem in order to prove the existence of φ . Claim 1: Fix any 0 > 0. Then, for all ∈ (0, 0 ) and φ ∈ Bρ (0) 2 kQδ,y ˜ δ,y ≤ CkφkHδ,y (φ)kH

(2.33)

and for any φ1 , φ2 ∈ Bρ (0) δ,y kQδ,y ˜ δ,y ≤ C(kφ1 kHδ,y + kφ2 kHδ,y )kφ1 − φ2 kHδ,y . (φ1 ) − Q (φ2 )kH

(2.34)


34

Proof of Claim 1: We have (see (2.28)), 2 |Qδ,y = 36|1 + f (x)|2 e8Uδ,y |e4φ − 1 − 4φ|2 (φ)|

≤ C|φ|4 e8(Uδ,y +|φ|) .

Using Lemma 2.2.1 we have 4 2 ωδ,y |Qδ,y (φ)|

≤C

kφk4Hδ,y (1 + log+ |x|)4 ec1 kφkHδ,y (1+log

+

|x|)

(δ 2 + |x − y|2 )4

which implies (2.33). Furthermore, δ,y 2 2 8Uδ,y 4φ1 |e − e4φ2 − 4(φ1 − φ2 )|2 |Qδ,y (φ1 ) − Q (φ2 )| = |1 + f (x)| e

(2.35)

and 4φ1

e

−e

4φ2

− 4(φ1 − φ2 ) = 16

Z

1

1

Z

4s(tφ1 +(1−t)φ2 )

e 0

0

ds(tφ1 +(1 − t)φ2 )dt (φ1 − φ2 ). (2.36)

Using (2.35) and (2.36) we have 4 δ,y 2 ωδ,y |Qδ,y ≤ Ckφ1 − φ2 k2Hδ,y ec1 (kφ1 kHδ,y +kφ2 kHδ,y )(1+log (φ1 ) − Q (φ2 )|

×

+

|x|)

(1 + log+ |x|)4 (kφ1 k2Hδ,y + kφ2 k2Hδ,y ) (δ 2 + |x − y|2 )4

and we get (2.34). Claim 2: For any compact set K ⊂ R+ × R4 and a ball Bρ (0) ⊂ Mδ,y with ρ > 0

small we can choose 0 = 0 (K, ρ) > 0 so that for any ∈ (0, 0 ), (δ, y) ∈ K,

the operator Gδ,y defined by (2.32) has a unique fixed point φδ,y ∈ Bρ (0) for all

∈ (0, 0 ). Moreover,

sup kφδ,y kHδ,y = O().

(2.37)

(δ,y)∈K

Proof of Claim 2 : Let (δ, y) ∈ K. For any φ ∈ Bρ (0), kGδ,y (φ)kHδ,y ≤ k((Sδ,y )0 (0))−1 k kSδ,y (0)kH˜ δ,y + kPR (Qδ,y ˜ δ,y . (φ))kH Now by Claim 1, there exists a constant C > 0 depending on K such that kGδ,y (φ)kHδ,y ≤ C[ + kφk2Hδ,y ], ∀(δ, y) ∈ K.

(2.38)


35

If we choose kφkHδ,y ≤ ρ where ρ is small enough and let 0 = (ρ − Cρ2 )/C, then for all ∈ (0, 0 )

kGδ,y (φ)kHδ,y ≤ ρ whenever kφkHδ,y ≤ ρ, ∀(δ, y) ∈ K. Now we show that Gδ,y is a contraction: δ,y kGδ,y (φ1 ) − Gδ,y (φ2 )kHδ,y ≤ k((Sδ,y )0 (0))−1 k k(Qδ,y ˜ δ,y (φ1 ) − Q (φ2 ))kH ≤ C(kφ1 kHδ,y + kφ2 kHδ,y )kφ1 − φ2 kHδ,y . Choosing φ1 , φ2 ∈ Bρ (0) with ρ small enough, we obtain Gδ,y : Bρ (0) → Bρ (0) is

a contraction map for all (δ, y) ∈ K and ∈ (0, 0 ). Hence by Banach fixed point theorem we obtain a unique fixed point φδ,y . Now, (2.37) follows from (2.38) by taking φ = φδ,y . This proves the claim. The proof of lemma follows from Claims 1 and 2.

2.4

Existence of Solution

First, we have the following technical fact: Proposition 2.4.1. Let φ ∈ Hδ,y . Define, Z ζ(R) = |x−y|=Rδ

−4 2 −2 2 wδ,y φ + wδ,y |∇φ|2 + |∆φ|2 + wδ,y |∇(∆φ)|2 dσ.

Then there exist a sequence of real numbers {Rn } with Rn → ∞ such that ζ(Rn ) = O(1) as n → ∞, Z (ii) |φ|dσ = o(Rn5 ) as n → ∞. (i)

|x−y|=Rn δ

Proof. We note that

R∞ 0

ζ(r)dr ≤ Ckφk2Hδ,y < ∞. Given any k > 0, let Ak = {r ∈

(0, ∞) : ζ(r) > k}. Clearly, k|Ak | ≤ Ckφk2Hδ,y . Therefore, by choosing k large enough, we can ensure |Ak | ≤ 1. Let Bk = (0, ∞) \ Ak . Then, it follows that ζ(r) ≤ k for all r ∈ Bk . We claim a stronger version of (ii) holds, viz., Z |x−y|=Rn δ

|φ|dσ = o(Rn5 ) as n → ∞ for any sequence {Rn } ⊂ Bk , Rn → ∞.


36

To prove this, we argue by contradiction i.e., suppose that there exist c, R0 > 0 such that for all R ∈ [R0 , ∞) ∩ Bk we get, Z

|φ|dσ ≥ cR5 > 0.

|x−y|=Rδ

(2.39)

By Hölder’s inequality, we obtain Z |x−y|=Rδ

|φ|dσ ≤

Z

4 dσ ωδ,y

21 Z

|x−y|=Rδ

|x−y|=Rδ

−4 ωδ,y |φ|2 dσ

12 .

(2.40)

But then, from (2.39) and (2.40), Z R4

−4 ωδ,y |φ|2 dx

= δ

−3

∞

Z

Z |x−y|=Rδ

0

≥ δ

−3

Z

Z

|x−y|=Rδ

[R0 ,∞)∩Bk

≥ O(1)


Z [R0 ,∞)∩Bk

dR


dR

1 dR = +∞, R

a contradiction. Hence (i), (ii) hold. The lemma below shows we can integrate by parts the functions in Hδ,y against (i)

ψδ,y . Lemma 2.4.2. Let φ ∈ Hδ,y . Then, for i = 0, 1, · · · 4, Z R4

(i) ψδ,y ∆2 φ

Z = 24 R4

(i)

e4Uδ,y ψδ,y φ.

Proof. We prove the lemma for i = 0, the cases i ≥ 1 are similar. As φ ∈ Hδ,y we obtain

Z R4

−4 ωδ,y |φ|2 dx

Z < +∞ and R4

|∆φ|2 < +∞.


37

Let the sequence {Rn } be as in the above proposition. Using (i), (ii) of this proposition, we deduce the following estimates: Z |x−y|=Rn δ

|φ|dσ = o(Rn5 ).

(2.41)

Z 12 Z 21 ∂φ −2 2 2 dσ ≤ ωδ,y |∇φ| dσ ωδ,y dσ |x−y|=Rn δ ∂ν |x−y|=Rn δ |x−y|=Rn δ

Z

7

Z |x−y|=Rn δ

≤ O(Rn2 ). Z 3 2 |∆φ|dσ ≤ O(Rn )

(2.42)

|x−y|=Rn δ

|∆φ|2 dσ

12

3

= O(Rn2 ).

(2.43)

12 Z 12 Z ∂∆φ 2 −2 dσ ≤ ∇(∆φ) 2 ωδ,y ωδ,y dσ dσ ∂ν |x−y|=Rn δ |x−y|=Rn δ |x−y|=Rn δ

Z

−1

≤ O(Rn 2 ).

(2.44)

Moreover, since φ ∈ Hδ,y , we obtain, Z R4

and

Z R4

(0) ψδ,y ∆2 φ

(0) ψδ,y e4Uδ,y φ

Z = lim

n→∞

|x−y|≤Rn δ

Z = lim

n→∞

|x−y|≤Rn δ

(0)

ψδ,y ∆2 φ

(0)

ψδ,y e4Uδ,y φ.


38

Using integration by parts, the last two equations and the above asymptotic estimates (2.41)-(2.44), we get Z |x−y|≤Rn δ

(0) ψδ,y ∆2 φ

Z

(0)

= 24 |x−y|≤Rn δ

Z + |x−y|=Rn δ

−

Z |x−y|=Rn δ

e4Uδ,y ψδ,y φ ! (0) ∂∆φ (0) ∂ψδ,y ψ − ∆φ dσ ∂ν δ,y ∂ν ! (0) ∂∆ψδ,y ∂φ (0) φ− ∆ψδ,y dσ ∂ν ∂ν

Z

(0)

e4Uδ,y ψδ,y φ |x−y|≤Rn δ Z |∆φ| ∂∆φ dσ + +O(1) Rn3 ∂ν |x−y|=Rn δ Z Z ∂φ −5 −4 dσ +O(Rn ) |φ|dσ + O(Rn ) |x−y|=Rn δ |x−y|=Rn δ ∂ν Z (0) = 24 eUδ,y ψδ,y φ + on (1).

= 24

|x−y|≤Rn δ

This proves the lemma. By the previous section, for any compact set K ⊂ R+ ×R4 , ρ > 0 small, there exists

0 > 0 such that for all ∈ (0, 0 ) and (δ, y) ∈ K, there exists φδ,y ∈ Bρ (0) ⊂ Mδ,y

such that Sδ,y (φδ,y ) = 0. For notational convenience, hereafter in this section we denote such a φδ,y simply as φ . Now we show that if (δ, y) is chosen carefully to be a stable zero of the vector field V0 , then for a sequence (δ , y ) → (δ, y), the function φδ ,y is in fact a zero of the

nonlinear operator Bδ ,y and hence

u = Uδ ,y + φδ ,y will solve (2.6). If φ ∈ Mδ,y solves Sδ,y (φ ) = 0, it follows that Bδ,y (φ ) ∈ R⊥ . Hence by Lemma

2.2.6, there exist constants ci, such that for all i = 0, 1, 2, 3, 4 Bδ,y (φ ) =

4 X i=0

(i)

ci, Φδ,y


39

and hence (i) hBδ,y (φ ), ψδ,y iL2 (R4 )

Z

(i)

= ci, R4

−4 ωδ,y |ψδ,y |2 ,

i = 0, 1, 2, 3, 4,

(2.45)

holds. Lemma 2.4.3. Let K ⊂ R+ × R4 be a compact set. If φ be obtained as in Lemma 2.3.1, then as → 0 we obtain for i = 0, 1, · · · , 4

(i) sup h∆2 φ − 6e4Uδ,y (e4φ − 1), ψδ,y iL2 (R4 ) = O(2 )

(δ,y)∈K

and (i) sup hf (x)(e4(Uδ,y +φ ) − e4Uδ,y ), ψδ,y iL2 (R4 ) = o (1).

(δ,y)∈K

Proof. Let K ⊂ R+ × R4 be a compact set and (δ, y) ∈ K. By (2.37), since φ → 0

0 in Hδ,y , we obtain φ → 0 in Cloc (R4 ). Using Lemma 2.4.2 and Theorem 2.2.1 we

obtain

Z R4

2

4Uδ,y

[∆ φ − 6e

(e

4φ

−

(i) 1)]ψδ,y

= −6 ≤

Z

(i)

e4Uδ,y [e4φ − 1 − 4φ ]ψδ,y

R4 Ckφ k2Hδ,y

= O(2 ).

Moreover, again by Theorem 2.2.1 and the dominated convergence theorem we get, hf (x)(e

4(Uδ,y +φ )

−e

4Uδ,y

(i) ), ψδ,y iL2 (R4 )

≤C

Z R4

Define the matrix Aδ,y = (Ai,j δ,y )0≤i,j≤4 by (i)

(j)

Ai,j δ,y = hΦδ,y , ψδ,y iL2 (R4 ) ; 0 ≤ i, j ≤ 4 and the vector



c0,



  c1,      c = c2,  .   c   3,  c4,

(i)

e4Uδ,y [eφ − 1]ψδ,y = o (1).


40

We note that Aδ,y is in fact an invertible diagonal matrix. Let K ⊂ R+ × R4 be a compact set with non-empty interior. Define the vector field

Z Z 1 (i) 4(Uδ,y +φ ) (i) 2 4Uδ,y 4φ f (x)e ψδ,y . V (δ, y) = (e − 1))ψδ,y − 6 (∆ φ − 6e R4 R4 i=0,1,··· ,4 Then from Lemma 2.4.3 we obtain V (δ, y) → 6V0 (δ, y) in C(K, R5 ). Now (2.45) can be written as

Aδ,y c = V (δ, y)

(2.46)

for (δ, y) ∈ K. Proof of Theorem 2.1.1. Let (δ, y) be a stable zero for the vector field V0 . Since V (δ, y) → 6V0 (δ, y) in C(K, R5 ), we may find zeroes (δ , y ) of V such that (δ , y ) → (δ, y). We can then take the solution φδ ,y of Sδ ,y (φ) = 0 given in

Lemma 2.3.1 and write out the corresponding equations (2.45) and (2.46) for Aδ ,y . Since Aδ ,y is invertible , we have c = 0 for all > 0. Hence the corresponding φδ ,y solves Bδ ,y (φδ ,y ) = 0 for all such . Defining u = Uδ ,y + φδ ,y ,

we obtain that u solves (2.6) for all > 0 small. That kφδ ,y kHδ,y = O() follows from Claim 2 in Lemma 2.3.1.

2.5

Necessary Condition

In this section we show that if there is a sequence of solutions u of (2.6) ”bifurcating” from some Uδ,y , then necessarily V0 (δ, y) = 0. The main tool to prove this

result is a Pohozaev type identity for functions belonging to Hδ,y . First, we prove the following sharp decay estimates: Lemma 2.5.1. Let u be a sequence of solutions of (2.6) with ||u − Uδ,y ||Hδ,y → 0

as → 0 for some (δ, y) ∈ R+ × R4 . Then, uniformly as → 0, we have the following decay estimates:

lim

u (x) = −2, |x|→∞ log |x|

(2.47)

lim x · ∇u = −2,

(2.48)

lim |x|2 |∆u (x)| = 4.

(2.49)

lim x · ∇(x · ∇u ) = 0.

(2.50)

|x|→∞

|x|→∞

|x|→∞


41

lim |x|2 x · ∇(∆u ) = 8.

(2.51)

|x|→∞

Proof. Let φ = u − Uδ,y . First note that kφ kHδ,y → 0 and by using Lemma 2.2.1

we deduce

|u − Uδ,y | 1 ≤ Ckφ kHδ,y 1 + →0 log |x| log |x|

(2.52)

as |x| → +∞. Using the fact that Uδ,y = −2, |x|→∞ log |x| lim

we obtain (2.47). We use similar arguments in [46] to establish (2.48), (2.49), (2.50) and (2.51). Using (2.47) we obtain ∀ 0 < ν < 2, ∃ R(ν) > 0 : u (x) ≤ (−2 + ν) log+ |x|, ∀ |x| > R(ν).

(2.53)

Then, since φ ∈ Hδ,y we can use (2.44) of Lemma 2.4.2 to conclude that for a suitable sequence Rn → ∞, Z 0

= lim

Rn →∞

∂BRn (0)

Z = lim

Rn →∞

BRn (0)

Z = lim

Rn →∞

BRn (0)

∂∆φ dσ = lim Rn →∞ ∂ν

Z BRn (0)

∆2 (u − Uδ,y )

6(1 + f (x))e4u − 6e4Uδ,y 6(1 + f (x))e4u − 16π 2 .

(2.54)

Hence, we obtain Z

∀ > 0,

(1 + f (x))e4u =

R4

8π 2 . 3

(2.55)

We define v by 1 v (x) = 2 8π

Z R4

log(|x − y|)6(1 + f (y))e4u (y) dy.

It is easy to check that ∆2 v = −6(1 + f (x))e4u in R4 and using (2.55) we obtain uniformly as → 0,

v (x) 3 lim = 2 |x|→∞ log |x| 4π

Z R4

(1 + f (y))e4u (y) dy = 2.

(2.56)


42

It can be shown, as in Lemma 2.2.1, that sup sup |v (x)| ≤ C(log+ |x| + 1).

0 0 and all |x| ≥ R = R(ν). The conclusions (i) and (ii) follow by differentiating

inside the integral sign in the definition of v .

We now develop two kinds of Pohozaev type identities. Lemma 2.5.3. Let {u } be a family of solutions to (2.6) such that ||u −Uδ,y ||Hδ,y → 0 as → 0 for some (δ, y) ∈ R+ × R4 . Then, Z R4

f (x)e4u

∂u = 0, ∂xi

i = 1, 2, 3, 4,

(2.57)

2. Q-Curvature Problem on S4 and

Z R4

43

f (x)e4u [(x − y) · ∇u + 1] = 0.

Proof. In order to prove (2.57) we multiply (2.6) by

∂u ∂xi

(2.58)

and integrate by parts on

the ball BR (0) to get Z

Z

4u ∂u

(1 + f (x))e

6

=

∂xi

BR (0)

∂BR (0)

∂∆u ∂u dσ − ∂ν ∂xi

Z BR (0)

∇ ∆u

∂u ·∇ . ∂xi (2.59)

By (2.51) and Corollary 2.5.2 (i), we obtain ∂∆u ∂u −1 ∂ν ∂xi dσ = O(R ) as R → ∞. ∂BR (0)

Z

(2.60)

Again,by suitable integration by parts and using (2.49) and Corollary 2.5.2 (ii), we get as R → ∞, Z BR (0)

∇ ∆u

∂u ·∇ ∂xi

1 ∂ ∂u 2 xi |∆u | dσ. = − ∆u ∂ν ∂xi 2R ∂BR (0) Z

= O(R−1 )

(2.61)

Hence, from the last two relations, lim {RHS of (2.59)} = 0.

(2.62)

R→∞

Again integrating by parts in another way, Z

4u ∂u

1 = ∂xi 4R

(1 + f )e BR (0)

Z xi e

4u

Z

f e4u

dσ + BR (0)

∂BR (0)

∂u . ∂xi

(2.63)

Using the asymptotic relation (2.47) and Corollary 2.5.2(i), we may let R → ∞ in the above equation to conclude Z lim

R→∞

4u ∂u

(1 + f )e BR (0)

∂xi

Z = R4

f e4u

∂u . ∂xi

(2.64)

Therefore we obtain,using (2.64) and (2.62), Z 6 R4

f (x)e4u

∂u = lim {LHS of (2.59)} = 0, ∂xi R→∞

(2.65)


44

which proves (2.57). Now we are left to show (2.58). For this, we multiply (2.6) by (x − y) · ∇u + 1 on either side and integrate on the ball BR (y) as before to obtain, Z e

6 BR (y)

4u

(1 + f (x)) ((x − y) · ∇u + 1) =

Z BR (y)

∆2 u ((x − y) · ∇u + 1). (2.66)

Integrating by parts we obtain, 3R LHS of (2.66) = 2

Z e

4u

Z dσ + 6

∂BR (y)

BR (y)

f e4u ((x − y) · ∇u + 1).

(2.67)

∂ = (x − y) · ∇. Again integrating by parts suitably, We denote r ∂r

1 ∂u ∂ 2 RHS of (2.66) = R |∆u | + +1 (∆u ) 2 ∂r ∂r ∂BR (y) ∂ ∂u −∆u r dσ. (2.68) ∂r ∂r Z

We have used the relation (obtained from integrating by parts) Z

R ∆u (x − y) · ∇(∆u ) = 2 BR (y)

Z

2

∂BR (y)

(∆u ) dσ − 2

Z

(∆u )2 dx

BR (y)

and the identity ∆((x − y) · ∇u ) = 2∆u + (x − y) · ∇(∆u ) to derive (2.68). Using the asymptotic (2.47)-(2.51), we obtain that, lim {LHS of (2.66)} = 6

R→∞

Z R4

f (x)e4u ((x − y) · ∇u + 1),

(2.69)

and lim {RHS of (2.66)} = 0.

(2.70)

R→∞

Hence (2.58) follows. Proof of Theorem 2.1.2. We note that (x − y) · ∇x Uδ,y + 1 = −δ

∂Uδ,y . ∂δ

Since u →

Uδ,y in Hδ,y , the asymptotics in lemma 2.5.1 allow us to pass to the limit as goes to 0 in (2.57) and (2.58). This means that V0 (δ, y) = 0.


2.6

45

Local Uniqueness of Solutions

In this section we show that a “strongly” stable zero of the vector field V0 (δ, y) “bifurcates” at most one family of solutions to (2.6).

Proof of Theorem 2.1.3. We argue by contradiction. Let us suppose that for some sequence n → 0 there exist two distinct sequence of solutions {u1,n } and {u2,n } of (2.6) such that ||ui,n − Uδ,y ||Hδ,y → 0 as n → ∞ for i = 1, 2. For convenience, we denote ui,n = ui,n . Set w ñ = u1,n − u2,n . Then ||wñ ||Hδ,y → 0 as n → ∞. Then, we have the following two cases: either Case(i): for any β > 0, for all large n, there exists xn ∈ R4 such that |wñ (xn )| ≥ β. or

Case(ii): There exists β > 0 and a subsequence of {wñ }, which we still denote by {wñ }, such that |wñ (x)| < β for all n and all x ∈ R4 . In this case, let xn ∈ R4 be such that |wñ (xn )| ≥ 21 ||wñ ||L∞ (R4 ) .

If Case(i) holds, we define: wn =

wñ , ||wñ ||Hδ,y

and if Case(ii) holds then wn =

wñ . kwñ kL∞ (R4 )

Then wn satisfies Z

2

∆ wn = 24(1 + n f (x))cn (x)wn with cn (x) =

1

e4tu1,n +(1−t)4u2,n dt. (2.71)

0

We note that, from (2.47), we have the estimate e4ui,n ≤ C(1 + |x|)ν−8 for any ν > 0, all |x| ≥ R = R(ν), and ∀n.

(2.72)

4 Using Schauder estimates, we obtain wn → w in Cloc (R4 ) where w satisfies the

problem

∆2 w = 24e4Uδ,y w

in R4 .

By nondegeneracy result in Lemma 2.2.2, w = c0

∂Uδ,y ∂δ

(2.73) +

∂Uδ,y i=1 ci ∂xi

P4

for some

ci ∈ R, i = 0, 1, · · · 4. We claim that ci = 0 for all i = 0, 1, · · · , 4. From the


46

identity (2.57) we get, Z

f (x)e4ui,n

R4

∂ui,n = 0, ∂xj

i = 1, 2; j = 1, 2, 3, 4.

(2.74)

Using assumption (2.12) and (2.72) we derive from (2.74) Z

∂f 4ui,n e = 0, ∂xj

R4

i = 1, 2 and j = 1, 2, 3, 4.

(2.75)

Therefore,

Z R4

∂f 4u2,n ∂f 4u1,n − e e ∂xj ∂xj

= 0 for j = 1, 2, 3, 4.

(2.76)

which can be written as Z R4

∂f cn (x)wn (x)dx = 0 for j = 1, 2, 3, 4. ∂xj

(2.77)

Using (2.12) we can pass to the limit in (2.77) to obtain, Z R4

4

∂f 4Uδ,y e ∂xj

∂Uδ,y X ∂Uδ,y + ci c0 ∂δ ∂xi i=1

! = 0,

j = 1, 2, 3, 4.

(2.78)

That is, integrating by parts again, 4 ∂ ∂Uδ,y X ∂Uδ,y 4Uδ,y f + ci e = 0, c0 ∂δ ∂x i R4 ∂xj i=1

Z

j = 1, 2, 3, 4.

(2.79)

Similarly, using (2.12) and (2.72) we deduce from (2.58), Z R4

h(x − y), ∇f ie4ui,n = 0 for i = 1, 2.

(2.80)

Then, arguing as above we get, Z R4

h(x − y), ∇f ie4Uδ,y w = 0.

Hence doing integration by parts we obtain that − 4δ

Z

4Uδ,y

f (x)e R4

∂Uδ,y w+ ∂δ

Z R4

f (x)e4Uδ,y h(x − y), ∇wi = 0.

(2.81)


47

Using the relations

∂w h(x − y), ∇wi = − δ +w , ∂δ and

Z

f (x)e4Uδ,y (x) w = 0

(from (2.78)),

R4

we rewrite (2.81) as − 4δ

Z f (x)e

4Uδ,y

R4

∂Uδ,y w−δ ∂δ

Z

f (x)e4Uδ,y

R4

∂w = 0. ∂δ

That is, 4 ∂Uδ,y X ∂Uδ,y ∂ 4Uδ,y c0 e + ci = 0. f (x) ∂δ ∂δ ∂x i R4 i=1

Z

(2.82)

Thus, from (2.79) and (2.82), we deduce D2 J(δ, y)c = 0 where c is the column vector (c0 , c1 , c2 , c3 , c4 )T . Since D2 J(δ, y) is an invertible matrix, we deduce c0 = 4 c1 = c2 = c3 = c4 = 0. This implies w ≡ 0 in R4 . Therefore, wn → 0 in Cloc (R4 )

and hence we necessarily have |xn | → ∞. Let us use the Kelvin transform to define

uî,n (x) = ui,n

x , |x|2

wˆn (x) = wn

Clearly, we have |wˆn ( |xxnn|2 )| ≥

1 2

x , |x|2

cˆn (x) = cn

x |x|2

x ∈ R4 \ {0}.

for all large n. It can be shown that wˆn satisfies

the following equation: 2

∆ wˆn

24 x = cˆn 1 + n f wˆn in R4 \ {0}. |x|8 |x|2

(2.83)

In Case(i), using the growth estimate (2.20), we get that |wˆn (x)| ≤ C(1 − log |x|)

4 (R4 \ {0}), by dominated for all n and all x ∈ B1 (0). Since wˆn → 0 in Cloc

convergence theorem we get that wˆn → 0 in Lp (B1 (0)) for all p ≥ 1. In Case(ii), 4 we have again, |wˆn | ≤ 1 and wˆn → 0 in Cloc (R4 \ {0}). Hence wˆn → 0 in Lp (B1 (0))

for any p ≥ 1. Using the assumption f ∈ L∞ (R4 ) and the estimate (2.72) we get that

24 x cˆn 1 + n f |x|8 |x|2

is a bounded sequence in Lp (B1 (0)) for any p > 1. Therefore the R.H.S in the equation (2.83) converges to 0 in Lp (B1 (0)) as n → ∞ for any p > 1. We recall

4 that wˆn → 0 in Cloc (R4 \ {0}). Using the standard Lp regularity theory (see for


48

example, Corollary 2.23 in [34]) and Sobolev embedding to the equation (2.83) we obtain ||wˆn ||L∞ (B1 (0)) → 0. This gives a contradiction easily in Case (i) and as well in Case (ii) since ||wˆn ||L∞ (B1 (0))

xn 1 ≥ wˆn ≥ |xn |2 2

for all large n. This proves the theorem.

2.7

Exact Multiplicity Result

Proof. Since the stable zeros of V0 are isolated there exists a R > 0 such that

zeroes of V0 are contained in the interior of a closed ball K = B R (0) ⊂ R+ × R4 .

Let m be the number of zeroes of V0 . By Theorems 2.1.1, 2.1.2 and 2.1.3 we

conclude that there exists 1 = 1 (K) > 0 such that for any ∈ (0, 1 ) the problem

(2.6) has at least m solutions ui and m points (δi , yi ) ∈ K such that ui − Uδi ,yi → 0 in Hδi ,yi , i = 1, ..., m. Let

Sµ = {u solves (2.6) for ∈ (0, µ), u − U1,0 ∈ H1,0 } \ {ui }0 0 small such that θµ ≥

θ0 2

for all µ < µ0 . By

Theorem 2.1.4, there exists some C > 0 and 2 > 0, d(ui , MK ) ≤ C,

i = 1, ..., m,

∈ (0, 2 ).


49

The conclusion of the theorem now follows by taking ρ0 = 0 = min{

2.8

θ0 2

and

θ0 , µ0 , 2 }. 2C

A Concrete Approach for Finding Stable Zeroes of V0

Throughout this section we assume (f 1)

f ∈ C 1 (R4 ) ∩ L∞ (R4 ).

By a change of variable J can be written as Z

f (δx + ξ) dx. (1 + |x|2 )4

J(δ, ξ) = 16 R4

(2.84)

Let Crit(f ), Crit(J) denote respectively the set of critical points of f and J. We have

Z J(0, ξ) = 16f (ξ) R4

1 dx. (1 + |x|2 )4

(2.85)

Since x 7→< ∇f (ξ), x > is an odd function, Z Dδ J(0, ξ) = lim (Dδ J) (δ, ξ) = 16 δ→0

R4

< ∇f (ξ), x > dx = 0. (1 + |x|2 )4

(2.86)

Therefore we can extend J as an even function of δ to R × R4 . Without loss of generality we denote this function by J. Also

ξ ∈ Crit(f ) ⇔ (0, ξ) ∈ Crit(J). Lemma 2.8.1. Assume the following conditions on f : (f2) there exists ρ > 0 such that < ∇f (x), x >< 0 for any |x| ≥ ρ, (f3) < ∇f (x), x >∈ L1 (R4 ),

R R4

< ∇f (x), x > dx < 0.


50

Then, there exists R > 0 such that < ∇J(δ, ξ), (δ, ξ) >< 0 whenever |(δ, ξ)| ≥ R.

(2.87)

Proof. See Lemma 3.3 in [3]. Remark 2.8.1. We note that the above Lemma 2.8.1 ensures that the set of critical points of J is compact. We make the following assumption about the “shape” of f near a critical point. (f4) Given ξ ∈ Crit(f ), suppose that there exists βξ = β > 1 such that :

(i) If β ≤ 4, there exist µ > 0 and a map Qξ : R4 → R, homogeneous of degree β, that is Qξ (λy) = λβ Qξ (y) for all y ∈ R4 , such that

f (y) = f (ξ) + Qξ (y − ξ) + O(|y − ξ|β+µ ) as y → ξ. (ii)If β > 4, we assume that f (y) = f (ξ) + O(|y − ξ|β ) as y → ξ. Lemma 2.8.2. Let (f4) hold. Then, as δ → 0+ ,     

δ β (Cβ,ξ + oδ (1)) 1 J(δ, ξ) − J(0, ξ) = 16 δ 4 log (C4,ξ + oδ (1))  δ    δ 4 (Cβ,ξ + oδ (1)) where

Cβ,ξ

 Z ∞ Z  rβ dr   Qξ (σ)dσ  2 )4  (1 + r 3  0 S  Z  = Qξ (σ)dσ  3  S  Z     |y|−8 [f (y + ξ) − f (ξ)]dy 

if β < 4, if β = 4,

(2.88)

if β > 4,

if β < 4, if β = 4,

(2.89)

if β > 4.

R4

Proof. Case 1 < β ≤ 4 : From (f 4)(i) we can find a C > 0 and 0 < R < 1 such

that

β β f (δx + ξ) − f (ξ) − δ |x| Qξ x ≤ C(δ|x|)β+µ , ∀|x| ≤ R . |x| δ

(2.90)

We remark that if β < 4 we can choose 0 < µ ˜ < µ small so that β + µ ˜ < 4. Since R < 1, we see that (2.90) still holds with µ ˜, which we continue to denote by µ.


51

We now compute, f (δx + ξ) − f (ξ) dx (1 + |x|2 )4 R4 Z f (δx + ξ) − f (ξ) dx = 16 (1 + |x|2 )4 B R (0) Zδ f (δx + ξ) − f (ξ) dx +16 (1 + |x|2 )4 R4 \B R (0)

J(δ, ξ) − J(0, ξ) = 16

Z

δ

= I

(1)

(δ) + I (2) (δ).

(2.91)

We simply estimate |I

(2)

Z

(δ)| ≤ 32kf k∞

R4 \B R (0) δ

1 dx = O(δ 4 ). (1 + |x|2 )4

(2.92)

Using (2.90) in the first integral I (1) (δ) we get |x|β Qξ

Z (1) β I (δ) − 16δ

B R (0) δ

x |x|

Z |x|β+µ β+µ ≤Cδ dx dx. 2 4 (1 + |x|2 )4 B R (0) (1 + |x| )

(2.93)

δ

If β < 4 (hence β + µ < 4), the above inequality gives I

(1)

(δ) = 16δ

β

Z

∞

0

rβ dr (1 + r2 )4

Z

Qξ (σ)dσ 1 + O(δ µ ) .

(2.94)

S3

If β = 4, again from (2.93) we get, I

(1)

1 (δ) = 16δ log δ 4

Z

Qξ (σ)dσ 1 + oδ (1) .

(2.95)

S3

Putting together (2.92),(2.94) and (2.95) we complete the case β ≤ 4. Case β > 4 : Using (f 4) and dominated convergence theorem, J(δ, ξ) − J(0, ξ) = 16δ

4

Z R4

|y|−8 (f (y + ξ) − f (ξ))dy + oδ (1).

This shows (2.88)-(2.89) for β > 1.

The proof of the following two results is a slight modification of Lemmas 3.6 and Lemma 3.8 respectively in [3].


52

Corollary 2.8.3. Let ξ ∈ Crit(f ) be isolated and assume that f satisfies (f 1) − (f 4). Suppose that Cβ,ξ 6= 0

(see (2.89)). Then q = (0, ξ) is an isolated critical

point of J and

Cβ,ξ > 0 =⇒ deg

loc (∇J, q)

= deg

loc (∇f, ξ)

Cβ,ξ < 0 =⇒ deg

loc (∇J, q)

= − deg

loc (∇f, ξ).

Proposition 2.8.4. If f has finitely many critical points and satisfies (i) assumptions (f 1) − (f 4) and Cβ,ξ 6= 0 (see (2.89)) and P (iii) degloc (∇f, ξ) 6= 1. (ii)

at any ξ ∈ Crit(f ),

Cβ,ξ 4 depends on global behavior of f , in contrast to the expressions for Cβ,ξ when β ≤ 4 which

depend of “shape” of f near ξ. It is easy to see that if ξ is a point of global maximum (minimum) for f, β = βξ > 4, then Cβ,ξ < 0 (resp > 0). Remark 2.8.3. In fact, if Crit(f ) ⊂ BR (0) for some R > 0 and for some suitably small we have maxx1 ,x2 ∈BR (0) |f (x1 ) − f (x2 )| < and minξ∈Crit(f ) |f (ξ)| > 1 , then

we can ensure that (ii) holds for all ξ ∈ Crit(f ) with β = βξ > 4 by letting f decay suitably outside the ball BR (0).

Remark 2.8.4. In the particular case, when β = 2, we obtain results similar to Wei-Xu [73], [74]. 2,µ Corollary 2.8.5. Let us suppose that f is a Cloc (R4 ) function satisfying:

(i) assumptions (f 1) − (f 4) at any ξ ∈ Crit(f ), (ii) for any ξ ∈ Crit(f ), ∆f (ξ) 6= 0 and, P (iii) degloc (∇f, ξ) 6= 1. ∆f (ξ) 0 such that (2.6) admits a solution u for all ∈ (0, 0 ). Moreover,

u = Uδ ,y + φ with φ → 0 in Hδ,y and (δ , y ) → (δ, y) as → 0. Furthermore, local uniqueness and exact multiplicity results as in Theorems 2.1.3, 2.1.4 hold if (δ, y) is a stable zero of J such that the Hessian D2 J(δ, y) is invertible and ∇f ∈ L∞ (RN ).

Chapter 3 Exact Multiplicity for the Perturbed Q-Curvature Problem in RN , N ≥ 5 3.1

Introduction

This chapter contains the study of local uniqueness and multiplicity of solutions to a fourth order equation with nonlinearity of critical exponent in RN , N ≥ 5. Let N ≥ 5 and D2,2 (RN ) denote the closure of C0∞ (RN ) in the norm kukD2,2 (RN ) := R ( |∆u|2 )1/2 . Let K ∈ C 2 (RN ). We consider the following problem for ε ≥ 0 : RN

(Pε )

 2,2 N    Find u ∈ D (R ) solving : N +4

∆2 u = (1 + εK(x))u N −4 ,

   u

> 0

) in RN .

We are interested in showing an exact multiplicity result for (Pε ) for all small ε > 0 (see Theorem 3.1.3 below). The above problem (P ) can be viewed as the analogue of the classical scalar curvature problem on (SN , g0 ). The above problem is a “perturbed”version of the well-known Q−curvature problem which arises in Differential Geometry. More precisely, the problem is to find out if a given smooth function Q on the N −dimensional

unit sphere SN is the Q−curvature function of a metric g on SN which is conformal 55

3. Q-Curvature Problem in RN , N ≥ 5

56

to the standard metric g0 . This gives rise to the following problem:

(P )

  Find v ∈ C 4 (SN ) solving:     ∆2 v − c ∆ v + d v = N g0 N g0  v >0     cN := 12 (N 2 − 2N − 4), dN :=

N +4 N −4 Qv N −4 2

1 N (N 16

) in SN

− 4)(N 2 − 4).

The above problem has been studied extensively using the background of differential geometry; see the works [32], [30] and [59] for the geometric context and references to other related works. ˜ for We now assume that Q is a perturbation of the constant, viz, Q = 1 + εK ˜ on SN and ε > 0 small. Then, applying the standard a smooth function K ˜ and v (and calling them K and u stereographic projection from SN to RN on K respectively), it can be checked that (P ) is transformed to (Pε ). Existence of solutions to (Pε ) on (SN , g0 ) for the above Q was done in [32] using a perturbative method (variational methods and finite dimensional reduction techniques) introduced in [1] and [2] and generalizing to the Branson-Paneitz operator the results of [3]. To describe their result, we make the following assumptions on K: (K1) K ∈ C 2 (RN ), kKkL∞ (RN ) + k∇KkL∞ (RN ;RN ) + kD2 KkL∞ (RN ;RN ×RN ) < ∞. (K2) (a) There exists ρ > 0 such that h∇K(x), xi < 0 ∀|x| ≥ ρ, R (b)h∇K(x), xi ∈ L1 (RN ), h∇K(x), xidx < 0. RN

(K3) The set of all critical points of K, denoted by crit (K), is finite. (K4) ∀ξ ∈ crit (K), there exists β = βξ ∈ (1, N ) and aj ∈ C(RN ), 1 ≤ j ≤ N

such that Aξ := Σj aj (ξ) 6= 0. Furthermore, K(y) = K(η) + Σj aj |y − η|β +

o(|x − y|β ) as y → η for any η in a small neighbourhood of ξ. (K5)

P Aξ 0 such that the following

hold: (i)

(Pε ) has a solution ∀ε ∈ (0, ε0 ),

(ii) (ref. [2]) ∀ε ∈ (0, ε0 ), there exists (µε , ξε ) ∈ IR+ ×RN such that for any compact

set A ⊂ IR+ ×RN , we may find a constant c(A) > 0 such that kuε −zµε ,ξε kD2,2 (RN ) ≤

c(A)ε.

(iii) (µε , ξε ) → (µ, ξ). We now define what we mean by a stable zero of a vector field.

3. Q-Curvature Problem in RN , N ≥ 5

58

Definition 3.1.2. Let G : IR+ × RN → IRN +1 be a C 1 vector field. We say that a point (δ, y) ∈ IR+ × RN is a stable zero for G if G(δ, y) = 0 and its derivative DG(δ, y) is an invertible matrix.

For a function u ∈ D2,2 (RN ), let d(u, MA ) = inf ku − zµ,ξ kD2,2 (RN ) . We can now (µ,ξ)∈A

state the following exact multiplicity result which we will prove in Section 3.4. Theorem 3.1.3. (ref. [62]) Let K satisfy the assumptions (K1) -(K5). We further suppose that V0 has finitely many zeroes in R+ × RN all of which are stable. Let

A ⊂ R+ × RN be any compact set containing the zeroes of V0 . Then there exists ρ0 = ρ0 (A) > 0 and = (ρ) > 0 such that for all ∈ (0, ), the problem (P ) has exactly the same number of solutions u with d(u, MA ) < ρ0 as the number of

zeroes of V0 .f

3.2

Preliminary Results.

We first prove a decay estimate (uniformly in > 0) for solutions of (P ). Theorem 3.2.1. Let {u }0 0 such that for any ∈ (0, 1 ) the

problem (P ) has at least M solutions, {ui }i=1 and M points {(µi , ξi )}M i=1 ⊂ A such that ui − zµi ,ξi → 0 in D2,2 (Rn ), i = 1, .., M, as → 0. For τ > 0 define Sτ = {u : u solves (P ) for ∈ (0, τ )}\{ui }0 0 such that θτ ≥

θ0 2

for all 0 < τ < τ0 . Also

from (ii) of Theorem 3.1.2, there exists some c > 0 and 2 > 0, d(ui , MA ) ≤ c, i = 1, 2, ...M, ∈ (0, 2 ). The theorem now follows by taking ρ0 =

θ0 2

and 0 = min

θ0 2

, τ0 , 2 .

Chapter 4 Isolated Singularities of Higher Order Elliptic Operators 4.1

Introduction

In this chapter we are interested in the qualitative analysis of solutions to semilinear elliptic equations or systems involving singular nonlinear terms. Singular problems arise in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrically conducting materials. The associated singular stationary or evolution equations give answer to various physical phenomena. Isolated singularities of elliptic operators are studied extensively, see for eg. [15], [48], [65], [67] and [68]. In this paper we wish to address the following problem and the questions related to it for the biharmonic(polyharmonic) operator in R4 (R2m ):Question: If a non negative measurable function u is known to solve a PDE in the sense of distribution in a punctured domain, then what can one say about the differential equation satisfied by u in the entire domain? In [15], Brezis and Lions answered this question for the Laplace operator with the assumption that 0 ≤ −∆u = f (u) in Ω \ {0} , u ≥ 0 , lim inf t→∞

69

f (t) > −∞ , Ω ⊂ RN . t

(4.1)

4. Isolated Singularities of Higher Order Elliptic Operator

70

With the above hypotheses it was proved that both u and f (u) belong to L1 (Ω), and satisfy −∆u = f (u) + αδ0 , for some α ≥ 0. For the dimension N ≥ 3, P.L.

Lions[48] found a sharp condition on f that determines whether α is zero or not in the previous expression. In [26], the authors further extended the result for dimension N = 2 by finding the minimal growth rate of the function f which guaranteed α to be 0. As it is well known, a singular solution of (4.1) is a function u belonging to some suitable functional class and such that there exists a sequence {xn } ⊂ Ω with xn → 0 and u(xn ) → ∞.

Taliaferro, in his series of papers (see for e.g. [67], [68], [35]) studied the isolated singularities of non-linear elliptic inequalities. In [68] the author studied the asymptotic behavior of the positive solution of the differential inequality 0 ≤ −∆u ≤ f (u)

(4.2)

in a punctured domain under various assumptions on f. If N ≥ 3 and the function f has a “super-critical”growth as in Lions[48], (i.e. limt→∞

f (t) N

t N −2

= ∞, ) then there

exists arbitrarily ’large solutions’ of (4.2). When N = 2, it was proved that there exists a punctured neighborhood of the origin such that (4.2) admits arbitrarily large solutions near the origin, provided that log f (t) has a superlinear growth at infinity. Moreover author characterizes the singularity at the origin of all solutions u of (4.2) when log f (t) has a sublinear growth. Later Taliaferro, Ghergu and Moradifam in [35] generalized these results to polyharmonic inequalities. The study of the polyharmonic equations of the type (−∆)m u = h(x, u) is associated to splitting the equation into a non-linear coupled system involving Laplace operator alone. Orsina and Ponce[56] proved the existence of solutions to

(1)

    −∆u =   

−∆v =

f (u, v) + µ

in Ω,

g(u, v) + η

in Ω,

u =v =0

on ∂Ω.

with the assumption that the continuous functions f and g are non increasing in first and second variables respectively with f (0, t) = g(s, 0) = 0. But here the authors assumed that µ and η are diffusive measures and Dirac distribution is not a diffusive measure. Considerable amount of existence/non-existence results have been proved for the problem (1) when f is a function of v alone and g depends only on u and µ, η are Radon measures. For example see [11] where the authors assumed


71

f (u, v) = v p , g(u, v) = uq and with non-homogeneous boundary condition. In [33] authors dealt with sign changing functions f and g, with a polynomial type growth at infinity and the measure µ and η were assumed to be multiples of δ0 . Our paper is closely related to the work of Soranzo [65] where author considers the equation: ∆2 u = |x|σ up with u > 0, −∆u > 0 in Ω ⊂ RN , N ≥ 4 and σ ∈ (−4, 0). A complete description of the singularity was provided when 1 < p
0 as n → ∞, we have Z

R2

R1

Thus,

Z Z 1 dr → 0. u (x)dS(x) − u(x)dS(x) n |SN |rN −1 |x|=r |x|=r

Z

un (x)dS(x) →

|x|=r

Z

u(x)dS(x) for a.e r ∈ (R1 , R2 ) .

|x|=r

Since R1 , R2 ∈ (0, R) are arbitrary , limn→∞ un (r) = u(r) for a.e r ∈ (0, R). Now

we show that

Z

R0

ψn (s)s r

1−N

ds →

Z r

R0

ψ(s)s1−N ds as n → ∞.


74

Indeed, Z

R0

ψn (s)s

Z

1−N

R0

s

= Z

un (t)dt ds

s R0

s

=

1−N

R0

Z s

r

Z

N −1

t

r

r

!

R0

Z

1−N

R0

= r

s1−N |SN |

1 |SN |

Z

|x|=t

Z

un (x)dS(x) dtds !

un (x)dx ds. BR0 \Bs

Therefore by dominated convergence theorem , Z lim

n→∞

R0

ψn (s)s1−N ds =

r

Z

R0

lim

n→∞

Z

R0

= r

Z

R0

= r

Z =

R0

r

1−N

s |SN |

1−N

s |SN |

s1−N |SN |

!

Z

un (x)dx ds BR0 \Bs

!

Z

un (x)dx ds

lim

n→∞

BR0 \Bs

!

Z

u(x)dx ds BR0 \Bs

s1−N ψ(s)ds.

r

Hence if u ∈ L1loc (BR \ {0}) we have for some constants C1 , C2 ≥ 0, u(r) ≤ a

Z

R0

r

ψ(s)s1−N ds + C1 r2−N + C2 , r ∈ (0, R).

(4.7)

For N = 2 we repeat the above computations by replacing r2−N by | log r|. In

particular, using the monotonicity of the functions s2−N and ψ we have from the last inequality, for N ≥ 3, rN −1 u(r) ≤ a

R R0 r

ψ(s)ds + C ≤ aR0 ψ(r) + C.

Again integrating the above inequality from r to R0 we see that ψ(r) ≤ a(R0 )2 ψ(r) + C for 0 < r ≤ R0 . Choosing R0 small enough so that a(R0 )2 < 1, we see

that ψ(r) remains bounded as r → 0 and consequently u ∈ L1loc (BR ). A similar

argument can be made for the case N = 2 also. Therefore from (4.7) we get u(r) ≤ C (ΓN (r) + 1)

for N ≥ 2.

(4.8)


75

Step 2: Define ϕ(x) = −∆u(x) a.e in BR . From the hypothesis we know that ϕ ∈ L1loc (BR \ {0}) ; we shall prove that in fact ϕ ∈ L1loc (BR ) . Define   N −2  N ≥ 3,  Φ ( ) |x| ξ (x) = log |x|   N = 2,  Φ log where Φ is a C ∞ convex function on [0,∞) such that Φ(0) = 1 and Φ(t) = 0 for t ≥ 1. It is clear from the definition that ξ is a C ∞ function on BR \ {0} and ξ (x) → 1 locally uniformly in BR \ {0}. It can also be seen that |∇ξ (x)| ≤ C|x|1−N . Hence |∇ξ | → 0 uniformly on compact subsets of BR \ {0} . Also

∆ξ ≥ 0 in BR \ {0} because for every > 0 , ξ = Φ ◦ g where Φ is convex and g is

harmonic in BR \ {0}. Let η ∈ C0∞ (BR ), 0 ≤ η(x) ≤ 1, η ≡ 1 in a neighbourhood of 0. Then,

R Ω

φξ η = −

R

= −

R

BR

Thus,

Z Ω

u(∆ξ )η − 2

R

u∇ξ .∇η − BR

R

BR

≤ −2

u∆(ξ η) BR

R

φξ η ≤ −2

Z BR

BR

u∇ξ .∇η −

R BR

uξ ∆η

uξ ∆η.

u∇ξ .∇η −

Z uξ ∆η.

(4.9)

BR

From the hypothesis of the theorem and the definition of ϕ we have ϕ + au − f ≥ 0 in BR . Since ξ and η are nonnegative, applying Fatou’s lemma we get, Z BR

lim inf (ϕ + au − f )ξ η dx =

Z

lim inf ϕξ η + (au − f )ηdx →0 Z Z (au − f )η. ≤ lim inf ϕξ η +

→0

BR

→0

Therefore

Z BR

Also lim→0

R BR

lim inf ϕξ η ≤ lim inf →0

→0

BR

BR

Z ϕξ η.

(4.10)

BR

u∇ξ .∇η = 0 because ∇η ≡ 0 near the origin and |∇ξ | → 0 as

→ 0 on compact subsets of BR \ {0}. Therefore taking lim inf →0 on both sides of R R (4.9) and using (4.10) we get BR ϕη ≤ − BR u∆η. Since, by the definition of ϕ, R R R (−au−f )η ≤ BR ϕη and −au−f ∈ L1loc (BR ) we obtain that BR (−au−f )η ≤ BR R R ϕη ≤ u∆η. This implies ϕ ∈ L1loc (BR ). BR BR


76

Step 3: Consider now the distribution T = −∆u − ϕ. The support of T is

contained in {0} and hence T = Σ|α|≤m cα Dα δ0 for some integer m ≥ 0. We claim that cα =0 when |α| ≥ 1. To show this claim take ξ ∈ C0∞ (RN ) such that

support(ξ) ⊂ BR and (−1)|α| (Dα ξ)(0) = cα for every |α| ≤ m. Let ξ (x) = ξ( x ), x ∈ RN . It is easy to see that −

Z

Z u∆ξ =

BR

X

ϕξ + BR

−|α| c2α .

(4.11)

|α|≤m

On the other hand we have Z

−2

Z

u∆ξ = BR

x u(x)∆ξ( )dx |x|≤R

and therefore, Z

|

−2

BR

u∆ξ | ≤ C

Z

−2

Z

u = C |x|≤R

u(r)rN −1 dr.

(0,R)

Using the estimate in (4.8) from Step 1 we get for some constants C1 , C2 > 0 |

R

|

R

BR BR

u∆ξ | ≤ C1

when

N ≥ 3,

u∆ξ | ≤ C1 |log| + C2

when

N = 2.

The above inequality combined with (4.11) gives that Σα≤m −|α| c2α ≤ C if N ≥ 3

which is only possible when cα =0 whenever |α| ≥ 1. And when N = 2 we get Σα≤m −|α| c2α ≤ C| log | + C. We note that for |α| = 6 0 −|α| → 0 faster than | log | as → 0 . So we conclude that cα = 0 for |α| ≥ 1. This proves our claim.

In conclusion we have proved that −∆u = ϕ + c0 δ0 in D0 (BR ) and ϕ ∈ L1loc (BR ).

Hence −∆(u − ϕ ∗ ΓN − c0 ΓN ) = 0 in D0 (BR ). Using Theorem 4.2.1 , we have u = h + ϕ ∗ ΓN + αΓN a.e in BR where h is a harmonic function. Therefore we get N/N −2

(BR ). Now we show that c0 ≥ 0. For that consider η ∈ C0∞ (BR ), 0 ≤ R R η ≤ 1 and η(x) ≡ 1 near x = 0. We have already shown that BR ϕη ≤ − BR u∆η R R and from the above conclusion we have BR ϕη = − BR u∆η − c0 , from which it u ∈ Mloc

follows that c0 ≥ 0. Thus the Theorem is proved.

Theorem 4.2.2. (Weyl Lemma, Simader[64]) Suppose G ⊂ RN be open and let u ∈ L1loc (G) satisfies Z G

u∆2 ϕdx = 0 for all ϕ ∈ Cc∞ (G), i.e. ∆2 u = 0 in D0 (G).


77

Then there exists u˜ ∈ C ∞ (G) with ∆2 u˜ = 0 and u = u˜ a.e in G. Theorem 4.2.3. (Weak maximum principle:) Let u ∈ W 4,r (Ω) be a solution of (

∆2 u = f (x) ≥ 0

in Ω

u ≥ 0, −∆u ≥ 0

on ∂Ω

Then we have u ≥ 0 and −∆u ≥ 0 in Ω. Proof of maximum principle easily follows by splitting the equation into a (coupled) system of second order PDE’s say: w = −∆u and −∆w = f with the

corresponding boundary conditions.

Using similar ideas we can in fact prove a maximum principle with weaker assumptions on the the smoothness of u, which is stated below: Theorem 4.2.4. Let u, ∆u ∈ L1 (Ω) and ∆2 u ≥ 0 in the sense of distributions.

Also assume that u, ∆u are continuous near ∂Ω and u > 0, −∆u > 0 near ∂Ω.

Then u(x) ≥ 0 in Ω.

Definition 4.2.1. Fundamental solution of ∆2 is defined as a locally integrable function Φ in RN for which ∆2 Φ = δ0 and precisely expressed as  1  |x|2 log |x|  8π    − 1 |x| 2w3 Φ(x) =  − w14 log |x|     1

2wN (N −2)(N −4)

if

4−N

|x|

N =2

if

N =3

if

N =4

if

N ≥ 5,

where wN is the surface measure of the unit sphere in RN . Theorem 4.2.5. Suppose g : Ω0 × [0, ∞) → IR+ be a measurable function and let u, ∆u and ∆2 u ∈ L1loc (Ω0 ). Let ∆2 u = g(x, u) in D0 (Ω0 ) with u ≥ 0 and −∆u ≥ 0 a.e in Ω0 . Then u, g(x, u) ∈ L1loc (Ω) and there exist a non-negative constants α , β

such that ∆2 u = g(x, u) + αδ0 − β∆δ0 in D0 (Ω).

Proof: Let us write w = −∆u. Then −∆w = g(x, u) ≥ 0 in D0 (Ω0 ) and also given

that w, g(x, u) ∈ L1loc (Ω0 ). Now as a direct application of Brezis-Lions Theorem 4.2.1, we obtain

− ∆w = g(x, u) + αδ0 for some α ≥ 0

(4.12)


78

and w, g(x, u) ∈ L1loc (Ω). Since −∆u = w ≥ 0 in Ω0 again by Theorem 4.2.1 u ∈ L1loc (Ω) and

−∆u = w + βδ0 for some β ≥ 0.

Now substituting w = −∆u − βδ0 in (4.12) we get ∆2 u = g(x, u) + αδ0 − β∆δ0 .

(4.13)

Extending g(x, u) to be zero outside Ω we get ∆2 (u − f (u) ∗ Φ − αΦ − βΓ) = 0

in D0 (Ω). By Weyl’s lemma for biharmonic operators, there exists a biharmonic function h ∈ C ∞ (Ω) and

u = g(x, u) ∗ Φ + αΦ + βΓ + h a.e in Ω. N

Note that Γ(x) belongs to Marcinkeiwicz space M N −2 (Ω) when N ≥ 2. By the N

property of the convolution of an L1 function with the functions in M N −2 (RN ) we N N −2 obtain u ∈ Mloc (Ω).

.

The above result has been proved in [65](see Theorem 2) as an application of their main result on the system of equations. Proof is essentially based on the idea of Brezis-Lions type estimates. We have instead given a direct alternative proof for the same result. Theorem 4.2.5 can be extended for polyharmonic operator in a standard way, for details see Theorem 4.4.1 .

4.3

Biharmonic Operator in R4

In this section we will restrict ourselves to the dimension N = 4 and g(x, u) to take a specific form g(x, u) = a(x)f (u). Let Ω be a bounded open set in R4 , 0 ∈ Ω 0

and denote Ω = Ω \ {0}. We assume

(H1) f : [0, ∞) −→ [0, ∞) is a continuous function which is non-decreasing in IR+ and f (0) = 0.

(H2) a(x) is a non-negative measurable function in Lk (Ω) for some k > 43 . (H3) There exists r0 > 0 such that essinf Br0 a(x) > 0.


79

Let u be a measurable function which solves the following problem: ( (P )

∆2 u = a(x)f (u) in Ω0 −∆u ≥ 0 in Ω0

u≥0 ,

From Theorem 4.2.5 we know that u is a distributional solution of (Pα,β )

(Pα,β )

 2    ∆u

= a(x)f (u) + αδ0 − β∆δ0

)

u ≥ 0 −∆u ≥ 0

in Ω,

   α, β ≥ 0, u and a(x)f (u) ∈ L1 (Ω).

The assumption (H3) suggests that the presence of such a weight function does not reduce the singularity of a(x)f (u) at origin. In particular, if a(x) = |x|σ for σ ∈ (−3, 0), then a(x) satisfies (H2) and (H3). Now assume that f (t) = c ∈ (0, ∞]. t→∞ t2 lim

(4.14)

i.e. f (t) grows at least at a rate of t2 near infinity. Then for some t0 large enough, c we have f (t) ≥ t2 for all t ≥ t0 . Suppose u is a solution of (Pα,β ) and f satisfies 2 4.14. Then we know that for some biharmonic function h u(x) = a(x)f (u) ∗ Φ + αΦ + βΓ + h a.e in Ω where Φ is the fundamental solution of biharmonic operator in R4 and Γ is the fundamental solution of −∆ in R4 . Since α and a(x)f (u) are non-negative, we have β u(x) ≥ β Γ(x) + h(x). If β 6= 0, fix an r˜ ∈ (0, r0 ) such that u(x) ≥ ≥ t0 2 2π |x|2 whenever |x| < r˜. Now, Z Br˜

a(x)f (u) ≥ C

Z Br˜

|x|−4 = ∞

which is a contradiction since a(x)f (u) ∈ L1 (Ω). Thus β = 0 if f (t) grows at a rate faster than t2 near infinity. We state this result in the next lemma.

Lemma 4.3.1. Let f satisfies the condition (4.14) and u solves (P ). Then for some α non-negative ∆2 u = a(x)f (u) + αδ0 in D0 (Ω). Now onwards we assume that f satisfies (4.14). We would like to address following questions in this paper:


80

1. Can we find a sharp condition on f that determines whether α = 0 or not in (Pα,0 )? 2. If α = 0, is it true that u is regular in Ω? Definition 4.3.1. We call f a sub-exponential type function if lim f (t)e−γt ≤ C

t→∞

for

some γ, C > 0.

We call f to be of super-exponential type if it is not a sub-exponential type function. We will show that the above two questions can be answered based on the nonlinearity being a sub-exponential type function or not. Theorem 4.3.1. (Removable Singularity)Let f be a super-exponential type function and u is a distributional solution of (P ). Then u extends as a distributional solution of (P0,0 ). Proof: Given u solves (P ), we know that (−∆)m u = a(x)f (u) + αδ0 − β∆δ0 for

some α, β ≥ 0. To show the extendability of the distributional solution we need to prove α = β = 0. Since f is of super exponential type function, from Lemma

4.3.1 it is clear that β = 0. Let us assume that α > 0 and derive a contradiction. α Note that we can find an r small enough such that u(x) ≥ − log |x| whenever 16π 2 |x| < r. Since f is not a sub-exponential type function, for a given γ > 0 there exists t0 > 0 such that f (t) ≥ eγt for all t ≥ t0 . Thus,

γα α f (u(x)) ≥ f − log |x| ≥ exp − log |x| , 16π 2 16π 2 for small x (that is |x| 0 and a C > 0, such that f (t) ≤ Ceγt for all t ∈ IR+ . Now define u(x) =

− log |x| + Cφ in B1 (0). γ

(4.15)

where φ is the unique solution of the following Navier boundary value problem,    (−∆)m φ = − a(x) log |x| in B1 (0) |x|   φ = 0 = ∆φ on ∂B1 (0).

(4.16)

We notice that since a(x) ∈ Lk (Ω), for some k > 34 , the term a(x)|x|−1 log |x| ∈

Lp (B1 ) for some p > 1. Hence the existence of a unique weak solution φ ∈ W 4,p (B1 )

is guaranteed by Gazzolla [34], Theorem 2.20. Now by maximum principle we have φ ≥ 0, −∆φ ≥ 0. Therefore,

u ≥ 0 in B1 (0),

− ∆u =

2 C − ∆φ ≥ 0. 2 γ|x| γ

(4.17)

(4.18)

and

∆2 u =

Note that a(x)f (u) ≤ C(Ω), and hence eCφ

δ0 C + a(x) |log |x|| . 2 8π γ γ|x|

(4.19)

C a(x)eCφ . By Sobolev embedding, we know W 4,p (Ω) ,→ |x| is bounded in B1 (0). Now we fix an r > 0 where eCφ ≤


82

| log |x|| in Br (0). We let Ω = Br (0) (where r depends only on γ and C) be γ C a(x)| log |x|| ≥ a(x)f (u). Now from the a strict subdomain of B1 (0) where γ|x| choice of r and equations (4.17), (4.18) and (4.19) it is obvious that u is a super 1 solution of (Pα,0 ) where α = 2 . Now let us define inductively with u0 = 0 8π γ

n (Pα,0 )

 2    ∆ un = a(x)f (un−1 ) + αδ0

in D0 (Ω)

  

on ∂Ω

un > 0, −∆un > 0

un = ∆un = 0

in Ω

Existence of such a sequence {un } can be obtained by writing un = wn + αΦ where  2    ∆ wn = a(x)f (un−1 ) in Ω,

wn = −αΦ, ∆wn = −α∆Φ on ∂Ω,    w ∈ W 4,r (Ω) for some r > 1. n Existence of w1 is clear since f (0) = 0 and from Theorem 2.2 of [34]. First let us show the positivity of u1 and −∆u1 in Ω. Since w1 is bounded, we can choose small enough so that u1 = w1 + αΦ > 0 and −∆u1 > 0 in B . In Ω \ B by weak

comparison principle we can show that u1 > 0 and −∆u1 > 0. Next we need to

show that u1 ≤ u. Note that by construction, u > 0 and −∆u > 0 in Br \ {0}. Then, u − u1 satisfies the set of equations (

∆2 (u − u1 ) ≥ 0

u − u1 > 0, −∆(u − u1 ) > 0

in D(Ω),

near ∂Ω.

Now using the maximum principle for distributional solutions (Theorem 4.2.4) we find u1 ≤ u. k Assume that there exists a function uk solving (Pα,0 ) for k = 1, 2 · · · n and

0 ≤ u1 ≤ u2 . . . ≤ un ≤ u in Ω. Since f is non-decreasing we have a(x)f (un ) ∈ Lp (Ω), for some p > 1. Thus by Sobolev embedding there exists a wn+1 ∈ C(Ω) ∩ W 4,p (Ω). Also, (

∆2 (un+1 − un ) = a(x)f (un ) − a(x)f (un−1 ) ≥ 0 un+1 = un , ∆un+1 = ∆un

in Ω on ∂Ω.


83

Again from weak comparison principle 0 < un ≤ un+1 and 0 ≤ −∆un ≤ −∆un+1 .

As before one can show that un+1 ≤ u. Now if we define u(x) = limn→∞ un (x) one can easily verify that u is a solution of (Pα,0 ) for α =

1 . 8π 2 γ

For a given f

sub-exponential type function we have found a ball of radius r such that (Pα,0 ) posed on Br (0) has a solution uα for α =

1 . 8π 2 γ

This solution uα is a supersolution

0

for (Pα0 ,0 ) posed in Br (0) and for α ∈ (0, α). Thus one can repeat the previous

iteration and show that for all α0 ∈ (0, α) there exists a weak solution for (Pα0 ,0 )

in Br (0).

Corollary 4.3.1. Suppose for a given γ > 0 there exists a Cγ such that f (t) ≤

Cγ eγt for all t ∈ IR+ . Then (Pα,0 ) has a solution in Brα (0) for all α ∈ (0, ∞). In δ

particular if f (t) = tp , p > 2 or et , δ < 1 then (Pα,0 ) is solvable for all α > 0.

Next we recall a Brezis-Merle [16] type of estimate for Biharmonic operator in R4 . Let h be a distributional solution of ( ∆2 h = f (2) h = ∆h = 0

in Ω on ∂Ω.

where Ω is a bounded domain in R4 . Theorem 4.3.4. (C.S Lin [46]) Let f ∈ L1 (Ω) and h is a distributional solution of

(2). For a given δ ∈ (0, 32π 2 ) there exists a constant Cδ > 0 such that the following inequality holds:

Z exp ( Ω

δh )dx ≤ Cδ (diamΩ)4 kf k1

where diam Ω denote the diameter of Ω. Theorem 4.3.5. Let f be a sub-exponential type function. Let u be a solution of (P0,0 ) with u = ∆u = 0 on ∂Ω. Then u is regular in Ω. Proof: Let u be a solution of ∆2 u = a(x)f (u) in Ω with Navier boundary conditions. Write g(x) = a(x)f (u), then g ∈ L1 (Ω). Fix a l > 0 and split g = g1 + g2 where kg1 k1
0. We use this Z

Choosing δ = 1 in Theorem 4.3.4, we find el|u1 |

exp(

higher integrability property of u in establishing its regularity. We can show that a(x)f (u) ∈ Lr (Ω) for some r > 1. In fact, Z

r

(a(x)f (u)) Ω

≤ C˜

Z

a(x)r eγru Ω 1/p0 1/p Z Z p0 γru pr e 1 close enough to 1 so that 1 < p.r ≤ k, where a(x) ∈ Lk (Ω).

Now let v be the unique weak solution of (

∆2 v = a(x)f (u) in Ω, v = 0, ∆v = 0 on ∂Ω.

0

We have v ∈ C 3,γ (Ω) for all γ 0 ∈ (0, 1). Now u = v + h for some biharmonic 0

function h. Therefore u ∈ C 3,γ (Ω).

.

Remark 4.3.1. The previous theorem is true even if a(x) ∈ Lk (Ω) for some k > 1. When f is super exponential in nature an arbitrary solution of ∆2 u = a(x)f (u) in D0 (Ω) need not be bounded. We consider the following example. 1

Example 4.1. Let w(x) = (−4 log |x|) µ for some µ > 1. Then one can verify that whenever x 6= 0,

µ

∆2 w = b1 ew w1−4µ [b2 w2µ − b3 ] µ

for some positive constants bi . Since f (w) = b1 ew w1−4µ [b2 w2µ − b3 ] is super ex-

ponential in nature, w extends as an unbounded distributional solution of ∆2 w = f (w) in Br (0) for r small enough.

4.4

Polyharmonic Operator in R2m

We suppose Ω is a bounded domain in RN , N ≥ 2m with smooth boundary and

0 ∈ Ω. We denote Ω0 as Ω \ {0}.

Theorem 4.4.1. Suppose g : Ω0 × [0, ∞) → IR+ is a measurable function and

∆k u ∈ L1loc (Ω0 ) for k = 0, 1, ..m. Let (−∆)m u = g(x, u) in D0 (Ω0 ) with (−∆)k u ≥ 0


85

for k = 0, 1, .., m − 1 a.e in Ω0 . Then u, g(x, u) ∈ L1loc (Ω) and there exist nonm−1 X m negative constants α0 , ..., αm−1 such that (−∆) u = g(x, u) + αi (−∆)i δ0 in i=0

D0 (Ω).

Now we restrict ourselves to dimension N = 2m and g(x, u) to take a specific form g(x, u) = a(x)f (u). Throughout this section we make the following assumption: (H10 ) f : [0, ∞) 7→ [0, ∞) is a continuous function which is non-decreasing in R+ and f (0) = 0.

(H20 ) a(x) is non negative measurable function in Lk (Ω) for some k >

2m . 2m−1

(H30 ) There exists r0 > 0 such that essinfBr0 a(x) > 0. Let u be a measurable function which satisfies the problem below,  m    (−∆) u = a(x)f (u)

(P 1 )

  

in Ω0

(−∆)k u ≥ 0 in Ω0 , k = 0, .., m − 1 u ∈ C 2m (Ω \ {0}).

Then by 4.4.1 we know that u is a distribution solution of (Pα10 ,..,αm−1 )

(Pα10 ,..,αm−1 )

          

(−∆)m u = a(x)f (u) +

m−1 X

αi (−∆)i δ0 in Ω

i=0

(−∆)k u ≥ 0, k = 0, .., m − 1 in Ω0

αi ≥ 0, for i = 0, .., m − 1 and u, a(x)f (u) ∈ L1 (Ω).

In [65], Soranzo et.al considered a specific equation (−∆)m u = |x|σ up in Ω0 , with

σ ∈ (−2m, 0) and (−∆)k u ≥ 0, for k = 0, 1, . . . m. By Corollary 1 of [65], if

N = 2m and p > max{1, N 2+σ } then α1 = α2 = · · · = αm−1 = 0 in (Pα10 ,...αm−1 ).

This result can be sharpened for any weight function a(x) satisfying (H3) in a standard way and we skip the details of the proof. f (t) = c ∈ (0, ∞]. Then we have t→∞ tm 1 = 0 in (Pα0 ,...,αm−1 ) and hence u is a distributional solution

Remark 4.4.1. Let u satisfy (P 1 ) and lim α1 = α2 = .. = αm−1

of (−∆)m u = a(x)f (u) + α0 δ0 in Ω. Now the following theorem gives us a sharp condition on f which determines α0 = 0 in (Pα10 ,0,...,0 ) and the proof is as similar to Theorem 4.3.1.


86

Theorem 4.4.2. Let f be a super-exponential type function and u is distribution 1 solution of (P 1 ). Then u extends as a distributional solution of (P0,0,..,0 ).

Theorem 4.4.3. Let f and a satisfy the hypotheses (H10 ) − (H30 ). Additionally f (t) assume lim m = c ∈ (0, ∞]. Then there exists an α0 > 0 such that for all t→∞ t 1 α ≤ α0 the problem (Pα,0,...0 ) admits a solution in Br (0), where the radius of the ball depends on the nonlinearity f .

Proof: We proceed as in Theorem 4.3.3, by constructing sub and super distri1 ) for all α small enough. We note that u0 = 0 is a butional solution for (Pα,0,..,0

sub-solution, and let u(x) =

− log |x| + Cφ in B1 (0) γ

(4.20)

where φ is the unique solution of the following Navier boundary value problem,    (−∆)m φ = − a(x) log |x| in B1 (0) |x|   φ = ∆φ = 0 = .. = (∆)m−1 φ on ∂B (0). 1

(4.21)

1 Then u is a supersolution of (Pα,0...0 ) in a small ball Br (0). Rest of the proof

follows exactly as in the case of biharmonic operator. Next we state a Brezis-Merle type of type of estimates for poly-harmonic operator in R2m . Theorem 4.4.4. (Martinazzi [52])Let f ∈ L1 (BR (x0 )), BR (x0 ) ⊂ R2m , and let v solve

(

(−∆)m v = f in BR (x0 ), v = (−∆)m v = ..... = ∆m−1 v = 0 on ∂BR (x0 )

Then, for any p ∈ (0,

γm kf kL1 (BR (x0 )) Z BR (x0 )

where γm =

), we have e2mp|v| ∈ L1 (BR (x0 )) and

e2mp|v| dx ≤ C(p)R2m ,

(2m − 1)! 2m S . 2

Finally with the help Theorem 4.4.4 we prove a regularity result for the polyharmonic operator.


87

Theorem 4.4.5. Let a(x) and f satisfies the properties as in (H10 ) − (H30 ) and 1 also assume that f be a sub-exponential type function. Let u be a solution (P0,0,..,0 ) 0

with u = ∆u = ... = ∆m−1 u = 0 on ∂Ω. Then u ∈ C 2m−1,γ (Ω), for all γ 0 ∈ (0, 1).

Chapter 5 Conclusion In this chapter we briefly recall the contents of this thesis and point out further directions of work suggested by some of the unanswered questions. This thesis was concerned with semilinear elliptic PDEs of higher order that arise in conformal geometry and singular problems. The first part deals with the so-called prescribed Q-curvature problem and the second part with isolated singularities of semilinear polyharmonic operators in even dimensions.

5.1

Q-Curvature Problem on the Standard Sphere

As described in the introduction, the prescribed Q-curvature (for definition see Chapter 1) problem asks: Given a smooth function Q on SN , N ≥ 4, does there exist a metric g conformal to the standard metric g0 such that Q = Qg ?

In order to answer the above question, we are led to the study of the following equations:   Pg40 u + 6 = 2Q(x)e4u 

PgN0 u =

N +4 N −4 Qu N −4 2

on S4 . (5.1) on SN , N > 4.

Here g0 denotes the standard metric on SN , N ≥ 4 and PgN0 denotes the corre-

sponding Panietz operator. Furthermore, Q is is assumed to be a perturbation of the constant. 89

5. Conclusion

90

In Chapter 2, we consider the problem in standard S4 and we prove existence results to the perturbed Q-curvature problem under assumptions only on the “shape” of Q near its critical points. These are more general than the non-degeneracy conditions assumed so far in the literature. We also show local uniqueness and exact multiplicity results for this problem. The main tool used is the Lyapunov–Schmidt reduction. In Chapter 3 we consider the perturbed Q-curvature problem in the standard sphere SN , N > 4. We prove a local uniqueness and exact multiplicity to the problem. The existence question to this problem had already been settled by [32]. For a further development in this direction we can take the manifold (M, g0 ) to be the hyperbolic space H4 with its standard metric g0 . We can similarly (as in [59]) look for a conformal metric g such that g has the prescribed Q-curvature. If we consider a conformal factor ρ = e2w then the Q-curvature, Qg with respect to the metric g is related to the Q-curvature Q0 with respect to given metric g0 via the equation: Pg0 + 2Q0 = 2Qg e4w in H4 .

(5.2)

We can look for the existence and multiplicity results for the equation (5.2). But the presence of exponential nonlinearity makes the probelm very difficult to handle. In this direction the existence of radial solutions for related Q-curvature problem in case of hyperbolic space HN , N > 4 has been studied in [38].

5.2

Isolated Singularities of Polyharmonic Operator

The second part of the thesis contained the study of isolated singularities of biharmonic operator with Navier boundary condition. In particular, we considered the following equation in a bounded domain Ω ⊂ R4 containing the origin: (P 0 )

∆2 u = g(x, u) ≥ 0 in D0 (Ω0 )

where Ω0 = Ω \ {0} with u ≥ 0 and −∆u ≥ 0 in Ω0 . Then it is known that u solves (P )

∆2 u = g(x, u) + αδ0 − β∆δ0 ,

5. Conclusion

91

for some constants α, β ≥ 0. In Chapter 4 we studied the existence of singular solutions to the problem (P ) in a small ball when α ≥ 0 and β = 0. We took g(x, u) = a(x)f (u) as our

model case, where a is a non-negative measurable function in a suitable Lebesgue

space. We connected the vanishing of α, β (seperately as well as simultaneously) to the growth of the nonlinear term f . Later, in the same chapter, we discussed analogous generalizations for the polyharmonic operator. Lots of questions remain unanswered in this direction. One is regarding the existence of a solution when α, β ≥ 0 in some suitable domain other than a ball.

Another major problem is the study of the equation (P 0 ) with the Dirichlet boundary condition on ∂Ω. The main obstacle in this situation is the lack of maximum principle in general domains.

Chapter 6 Appendix 6.1

Introduction

In this chapter we recall some well-known concepts and results required in the thesis. We give an overview of the topics finite dimensional reduction and Kelvin transformation. The finite dimensional reduction technique or Lyapunov-Schmidt method is a useful tool when implicit function theorem fails. It allows us to do the reduction of equations in infinite-dimensional Banach spaces to equations in finitedimensional spaces. Later we prove a useful transform for harmonic functions, sometimes called the Kelvin transform.

6.2

Lyapunov-Schmidt (or Finite Dimensional) Reduction

The Lyapunov-Schmidt method is essentially a reduction procedure by which a PDE operator equation in infinite dimensional function spaces is reduced to an equivalent equation in finite dimensions. This is done by making use of a splitting of the solution space into subspaces of the Range and Kernel type. The method was discovered in the early twentieth century by A.M Lyapunov [50] and E. Schmidt [63] through their studies on nonlinear integral equations (for more details see [70]).

93

6. Appendix

94

The reduction technique has been used in a large variety of problems: continuous and discrete systems, ordinary, partial and functional differential equations, integral equations, variational problems etc. More specifically, the Lyapunov-Schmidt reduction has been applied to local bifurcation problems, existence of periodic solutions and the global existence and multiplicity results for solutions of infinitedimensional problems. The common theme in each of these applications is that of a reduction: the original equation is partly solved, and substitution of the partial solution in the remaining equations gives a reduced equation of a lower dimension than the original one.

6.2.1

An Overview of The Reduction Method in Infinite Dimensions

Solving nonlinear bifurcation problems in partial differential equations generally leads us to equivalent operator equations in infinite dimensional function spaces. We sketch a basic abstract setting, which under appropriate conditions, allows us to reduce these infinite dimensional operator equations into finite dimensional ones. Let X, Y be Banach spaces and consider a C 1 map (A.1)

F : R × X → Y such that F (λ, 0) = 0 for all λ ∈ IR.

Let us call the set {(λ, 0) : λ ∈ IR} as the set of trivial solutions. We now wish to find non-trivial solutions to the equation

F (λ, x) = 0, (λ, x) in a neighbourhood of (0, 0).

(6.1)

Let us denote the set of non-trivial solutions by ΣF = {(λ, x) ∈ R × X : x 6= 0, F (λ, x) = 0}.

(6.2)

Clearly, if Dx F (λ, 0) is invertible for every λ ∈ IR, by implicit function theorem

there is no hope of obtaining a non-trivial solution to (6.1). Therefore, necessarily we need to assume that (A.2)

∃ λ∗ ∈ IR be such that L := ∂x F (λ∗ , 0) is not invertible.

6. Appendix

95

Let us denote X1 = Kernel(L) and Y2 = Range(L). The following hypotheses on the operator L are crucial for the Lyapunov-Schmidt reduction: (L1) X1 is finite dimensional; (L2) the range space Y2 is closed and has finite codimension in Y . (L3) L is a Fredholm operator i.e dim(X1 ) = codim(Y2 ). However, it is well-known that finite-dimensional and finite-codimensional closed subspaces can be complemented. Therefore, we may find closed subspaces X2 ⊂ X

and Y1 ⊂ Y such that the following decompositions hold: X = X1 ⊕ X2 ,

Y = Y1 ⊕ Y2 .

Thus for any x ∈ X there exist unique x1 ∈ X1 and x2 ∈ X2 such that x = x1 + x2 .

Let P1 and P2 denote respectively the projections of Y onto Y1 and Y2 . We then

see that any solution x = x1 + x2 of (6.1) gives rise to a solution of the following system and vice-versa: P1 F (λ, x1 + x2 ) = 0

(6.3)

P2 F (λ, x1 + x2 ) = 0

(6.4)

The above equation (6.4) is sometimes called the auxiliary equation. Y2

X2

Y = Y1 ⊕ Y2

X = X1 ⊕ X2 F (λ, ·) x2 b

P2 P2 y

bb

b

b

y = P1 y + P2 y

x = x1 + x2 P1 b

b

0

b

x1

X1

Figure 6.1

We can prove the following lemma

b

0

P1 y

Y1

6. Appendix

96

Lemma 6.2.1. Let the assumptions (A.1),(A.2) and (L1)-(L3) hold. Then, the auxiliary equation (6.4) is uniquely solvable in a neighbourhood of (λ∗ , 0). More precisely, there exist neighbourhoods (λ∗ − , λ∗ + ) of λ∗ , B1 (0) of x1 = 0 in X1 ,

B2 (0) of x2 = 0 in X2 , and a map f2 ∈ C 2 ((λ∗ − , λ∗ + ) × B1 (0), X2 ) such that P2 F (λ, x1 + x2 ) = 0, for some (λ, x1 , x2 ) ∈ U ⇔ x2 = f2 (λ, x1 ) where U := (λ∗ − , λ∗ + ) × B1 (0) × B2 (0). Furthermore, we also have f2 (λ, 0) = 0, ∂x2 f2 (λ∗ , 0) = 0.

∀λ ∈ (λ∗ − , λ∗ + )

(6.5) (6.6)

Proof. We define φ ∈ C 1 (R × X1 × X2 , Y2 ) by φ(λ, x1 , x2 ) = P2 F (λ, x1 + x2 ).

Then ∂x2 φ(λ∗ , 0, 0) is the linear map from X2 to Y2 , which is the restriction of L to X2 . Indeed, for x2 ∈ X2 , ∂x2 φ(λ∗ , 0, 0)(x2 ) = P2 ∂x F (λ∗ , 0)(x2 ) = P2 Lx2 = Lx2 . Hence ∂x2 φ(λ∗ , 0, 0) is a continuous linear isomorphism from X2 onto Y2 . Since Y2 is closed, it follows that ∂x2 φ(λ∗ , 0, 0) is a linear homeomorphism between these two spaces and then one can apply Implicit function theorem to characterise the solutions of (6.4) in the neighbourhood of (λ∗ , 0). We now substitute x2 = f2 (λ, x1 ) into (6.3) and we obtain the bifurcation equation: G(λ, x1 ) := P1 F (λ, x1 + f2 (λ, x1 )) = 0,

x1 ∈ X1 .

(6.7)

Notice that the above equation is a finite dimensional equation. To recap, To solve (6.1) we need to solve for each λ 6= λ∗ , the the finite set of k equations in k variables (k = dim(X1 ) = dim(Y1 )) given by (6.7).

6. Appendix

97

We note that the bifurcation mapping G : R × X1 → Y1 is smooth, with G(0, 0) = 0, ∂x1 G(0, 0) = 0 and ∂λ G(0, 0) = ∂λ P1 F (0, 0). Further analysis of the bifurcation equation (6.7) will of course strongly depend on the details of the problem such as dimensions of X1 and Y1 , requirements on partial derivatives of G (as in Crandall-Rabinowitz bifurcation theorem), possible symmetries etc. The main difficulty in analyzing the equation (6.7) consists in the fact that typically the solution f2 (λ, x1 ) of the equation (6.4) is not known explicitly, and hence also the (bifurcation) mapping G. It is therefore important that projections P1 and P2 used in the reduction should be such that at least some qualitative properties of f2 and G can be obtained. When the method is used to study the local bifurcation problems also a good approximation of these mappings, such as for example a few terms in there Taylor expansion, is required to obtain relevant conclusions. Assume that there exists a sequence of solutions (λn , x1,n ) of (6.7) such that

(λn , x1,n ) → (λ∗ , 0) with x1,n 6= 0. From the above discussions, if we set xn = x1,n + x2 (λn , x1,n ), we obtain

F (λn , xn ) = 0. Now from the smoothness of the map f2 and (6.5) we have f2 (λn , x1,n ) → 0.

Finally,

if x1,n 6= 0 then xn := x1,n + f2 (λn , x1,n ) 6= 0, which implies that (λn , xn ) is a nontrivial solution of (6.1). From the preceding discussion what we have proved is the following theorem for one-parameter bifurcation: Theorem 6.2.1. Let F ∈ C 1 (R × X, Y ) satisfy assumptions (A.1),(A.2),(L1)(L3). Suppose that the bifurcation equation (6.7) possesses a sequence of solutions

(λn , x1,n ) → (λ∗ , 0) with x1,n 6= 0. Then, setting xn = x1,n + f2 (λn , x1,n ), one has

that (λn , xn ) ∈ ΣF , xn → 0 and thus λ∗ is a bifurcation point for a branch of non-trivial solutions to (6.1).

6. Appendix

6.3

98

Kelvin Transformation

The Kelvin Transformation is a tool that helps us to study the behavior of harmonic functions at infinity. The usefulness of this technique also extends to the study of subharmonic, superharmonic and polyharmonic functions. This transformation technique seems to have been introduced by W. Thomson (Lord Kelvin, [69]). For more details see [77], [41]. We start with the concept of inversion with respect to a sphere. With out loss of generality we can take a sphere SR (0) centered at the origin of radius R. We define the inversion of a point x ∈ Rn with respect to the sphere SR (0) as x∗ :=

R2 x. |x|2

It is easy to see that under this inversion 0∗ = ∞, ∞∗ = 0 and (x∗ )∗ = x. Furthermore, under this inversion Sr (0) transforms into SR2 /r (0), the interior of a sphere to is transformed to the exterior and vice versa. Therefore, Kelvin transformation enables one to reduce exterior problems in potential theory to interior ones and the vice versa (see [71]). Definition 6.3.1. (Kelvin Transform)Let Ω ⊂ RN be an open set which does not

contain 0 and denote by Ω∗ the image of Ω under the Kelvin transform with respect to the sphere SR (0). Then for any function u : Ω → R, the Kelvin transform u∗

of u is given by

|x|N −2 1 1 u (x ) = 2N −4 u(x) = ∗ N −2 u(x) = ∗ N −2 u R |x | |x | ∗

∗

R2 ∗ x , x∗ ∈ Ω∗ . |x∗ |2

The nice behaviour of Kelvin transform with respect to polyharmonic operator is stated in the following theorem. Theorem 6.3.1. (Lemma A.3 of [41]) Given u ∈ C ∞ (RN ) define u∗ (x) := u

x |x|2

for x ∈ RN \ {0}. Then for any k ∈ N we have k

∆

1 x k u (x) = N +2k (∆ u) , |x|N −2k |x| |x|2 1

∗

x ∈ RN \ {0}

(6.8)

6. Appendix

99

Proof. We prove using the method of induction. Step 1: We note that the (6.8) is true for k = 0. Step 2: Now we assume that (6.8) is true for some k ≥ 0. We can check that for any smooth function f and g(x) := |x|2 , we have

∆k+1 (f g) = g∆k+1 f + 2(k + 1)(N + 2k)∆k f + 4(k + 1)x · ∇(∆k f ). We compute k+1

∆

u∗ (x) |x|N −2(k+1)

= ∆k+1 (f g) = g∆(∆k f ) + 2(k + 1)(N + 2k)∆k + 4(k + 1)x · ∇(∆k f ) 1 x 2 k = |x| ∆ (∆ u) |x|N +2k |x|2 x 2(k + 1)(N + 2k) k (∆ u) + |x|N +2k |x|2 1 x k + 4(k + 1)x · ∇ (∆ u) |x|N +2k |x|2 x 1 . = N +2(k+1) (∆k+1 u) |x| |x|2

That is, (6.8) is true for k + 1 whenever it’s true for k. In view of the above theorem we can state Theorem 6.3.2. Let Ω ⊂ RN be an open set which does not contain 0. Then a function u is polyharmonic (that is ∆k u = 0 for some integer k ≥ 2) in Ω if and only if the Kelvin transform u∗ is polyharmonic in Ω∗ .

6.4

Marcinkiewicz Spaces

Marcinkiewicz spaces are also known as Weak-Lp , was introduced by J. Marcinkiewicz. One of the real attraction of Weak-Lp space is that the subject is sufficiently concrete and yet the spaces have fine structure of importance for applications. WeakLp spaces are function spaces which are closely related to Lp spaces. The Book by Colin Benett and Robert Sharpley [9] contains a good presentation of Weak-Lp spaces.

6. Appendix

100

Definition 6.4.1. (Distribution Function) Given a measurable function u on Ω ⊂

RN , N ≥ 2, we associate with it the distribution function u∗ defined as below: u∗ (α) = |{x ∈ Ω; |u(x)| > α}| , α > 0, where |E| denotes the N dimensional Lebesgue measure of the set E.

It is immediate to note that the distribution function u∗ is a non-increasing function which is continuous from right. Definition 6.4.2. (Marcinkiewicz Space) For p ≥ 1, we define the Marcinkiewicz space M p (Ω) as the class of all measurable functions u on Ω such that [u]M p (Ω) = sup α(u∗ (α))1/p

(6.9)

α>0

is finite. M p (Ω) is also sometimes called the weak -Lp space and denoted by Lp∗ (Ω). From the so called Chebyschev inequality, Z

p

|u(x)| dx ≥

Z {x;|u(x)|>α}

|u(x)|p dx ≥ αp u∗ (α),

we obtain [u]M p (Ω) ≤ ||u||Lp (Ω) . Clearly Lp (Ω) ⊂ M p (Ω). The functional u → [u]M p (Ω) does not give a norm on M p (Ω). We will describe below briefly how to

construct a suitable norm on M p (Ω). We note that given a measurable set K ⊂ Ω, the characteristic function χK satisfies the relation

[χK ]M p (Ω) = sup(α |{x : |χK (x)| > α}|)1/p = |K|1/p = ||χK ||Lp (Ω) . α>0

Proposition 6.4.1. For all 1 ≤ q < p < ∞, M p (Ω) ⊂ Lqloc (Ω). Proof. To prove the proposition it suffices to prove the following inequality: Z K

|u(x)|dx ≤

p−1 p |K| p [u]M p (Ω) p−1

(6.10)

holds for every measurable K ⊂ Ω. Then, indeed the proposition follows from the fact u ∈ M p (Ω) iff |u|q ∈ M p/q (Ω). We can always assume that 0 < |K| < ∞ and

6. Appendix

101

0 < [u]M p (Ω) < ∞. We can show without difficulty that (uχK )∗ (α) ≤ inf{|K|, u∗ (α)}

(6.11)

Let A = {x : |u(x)|.|χK (x)| > α} and B = {x : |u(x)| > α}. Then clearly A ⊂ B and A ⊂ K from which (6.11) follows. Hence, for 0 < r < ∞ : Z K

|u(x)|dx = ≤

∞

Z

Z0 r

(uχK )∗ (α)dα Z ∞ u∗ (α)dα |K|dx + r

0

≤ |K|r +

[u]pM p (Ω)

r1−p . p−1

Minimizing over 0 < r < ∞ we get , Z K

|u(x)|dx ≤

p−1 p |K| p [u]M p (Ω) . p−1

This proves the proposition. The inequality (6.10) also allows us to show that the Marcinkiewicz space M p (Ω) is normable. Indeed, for p > 1 , put ||u||M p (Ω) =

|K|

sup K⊂Ω,|K| α}, 1/p

α|K|

1/p−1

≤ |K|

Z K

|u(x)|dx ≤ ||u||M p (Ω)

It follows that u ∈ M p (Ω) iff ||u||M p (Ω) < ∞. It is also immediate that ||.||M p (Ω) is a norm on M p (Ω) .

Proposition 6.4.2. (Young’s inequality): If a ∈ L1 (Ω) and b ∈ M p (Ω) then a ∗ b ∈ M p (Ω) and furthermore ||a ∗ b||M p (Ω) ≤ ||a||L1 (Ω) ||b||M p (Ω) .

6. Appendix

102

Proof. Define a and b to be zero outside Ω . Let K ⊂ Ω be any set of finite measure. Then,

Z Z dx |(a ∗ b)(x)|dx = a(y)b(x − y)dy K K RN Z Z |a(y)|dy |b(x − y)|dx ≤ N R K Z p−1 ≤ |a(y)|.||b||M p (Ω) |K − y| p dy

Z

RN

= ||a||L1 (Ω) |K|

p−1 p

||b||M p (Ω) .

Now by taking supremum over all such K and from equation (6.12) we get ||a ∗ b||M p (Ω) ≤ ||a||L1 (Ω) ||b||M p (Ω) . Example 6.4.1. Take the fundamental solution of −∆ in RN : ( ΓN (x) =

where CN =

1 , ωN N (N −2)ωN

CN |x|2−N N ≥ 3, 1 2π

1 log( |x| ) N = 2,

= |S N −1 |. N

Then for N ≥ 3 , ΓN 6∈ L N −2 (Ω) for any bounded open set containing origin in

RN . But we can easily check that ΓN ∈ M N/N −2 (RN ) because for every α > 0 we

have

αΓ∗ (α)

N −2 N

N −2 = α {x : CN |x|2−N > α} N N −2 1 N N −2 N −2 −1 = β {x : C |x| > β} N

where β = α1/N −2 . Remark 6.4.1. Using the propositions 6.4.1 and 6.4.2 we note that if we convolve any u ∈ L1 (Ω) with the fundamental solution ΓN , the resulting function u ∗ ΓN

will be in M N/N −2 (Ω) and hence in Lploc (Ω) for every p

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