Non-radial Maximizers For Functionals With Exponential Non-linearity

Advanced Nonlinear Studies 5 (2005), 337–350

Non-radial Maximizers For Functionals With Exponential Non-linearity in R2 Marta Calanchi, Elide Terraneo Dipartimento di Matematica ”F. Enriques” Universit` a di Milano, via Saldini, 50, 20133 Milano, Italy e-mail: [email protected], [email protected] Received 1 October 2004 Communicated by Jean Mawhin

Abstract We consider the functional F : H01 (B(0, 1)) → R Z γ |x|α (ep|u| − 1 − p|u|γ ) dx F (u) = B(0,1)

where α > 0, p > 0, 1 < γ ≤ 2, and B(0, 1) is the unit ball in R2 . We prove that for any p > 0, 1 < γ < 2 and 0 < p < 4π, γ = 2 no maximizer of F (u) on the unit ball in H01 is radially symmetric provided that α is large enough. This extends a result of Smets, Su and Willem concerning the existence of non-radial ground state solutions for the Rayleigh quotient related to the H´enon equation with Dirichlet boundary conditions. 2000 Mathematics Subject Classification. 35J60. Key words. non-radial solutions, exponential growth.

1

Introduction

The present work is mainly motivated by a paper due to Smets, Su, Willem (see [14]) in which the authors consider the problem of the symmetry of positive solutions for the H´enon equation with Dirichlet boundary conditions 337

338

M. Calanchi, E. Terraneo

 α p−1 in B(0, 1),  −∆u = |x| u u > 0 in B(0, 1),  u|∂B(0,1) = 0,

(1.1)

where B(0, 1) denotes the unit ball in RN , N ≥ 2, centered at the origin, α > 0, p > 2. The existence of solutions in the Sobolev space H01 for N ≥ 3 was first considered by Ni in 1982 [11]. He realizes that the presence of the weight |x|α widens the range of p for which a solution exists. Ni proves via an application of the Mountain Pass Theorem in the space of radial functions in H01 that problem (1.1) possesses a radial solution for any 2N ∗ p ∈ (2, 2∗ + N2α −2 ), where 2 = N −2 is the critical exponent of the Sobolev embedding if N ≥ 3. Moreover thanks to the Pohozaev identity, one can show that there are no solution if p ≥ 2∗ + N2α −2 . In contrast with the case in which the weight is a non-increasing radial function (see [8]), (1.1) has not only radial solutions. Indeed in [14] Smets, Su, Willem prove that for any p ∈ (2, 2∗ ), there exists at least a non-radial solution of (1.1) provided that α is large enough. This result also holds in the case of dimension N = 2 where 2∗ = +∞. They study the ground state solutions of (1.1), i.e. functions which minimize the Rayleigh quotient R |∇u|2 dx B(0,1) p Rα (u) =  u ∈ H01 , u 6= 0. (1.2)  p2 , R α |u|p dx |x| B(0,1) Under suitable rescalings, the minimizers of (1.2) are solutions of (1.1). Then they establish that for α sufficiently large the infimum of (1.2) is attained by a non-radial function. Theorem [Smets, Su, Willem [14]]. Assume N ≥ 2. For any p ∈ (2, 2∗ ) there exists α∗ such that any minimizer of (1.2) is non-radial provided α > α∗ . They also give some information about the behaviour of α∗ as p goes to 2 or to 2∗ . In particular they show that α∗ goes to 0 when p tends to 2∗ and α∗ goes to +∞ if p tends to 2. In [13] Serra investigates the existence of non-radial solutions to (1.1) in the critical case p = 2∗ when the spatial dimension N is greater than or equal to 4. He proves the following theorem: Theorem [Serra [13]]. Let N ≥ 4. Then for every α > 0 large enough, the problem  α 2∗ −1 in B(0, 1),  −∆u = |x| u u > 0 in B(0, 1),  u|∂B(0,1) = 0, admits at least one non-radial solution.

(1.3)

339

Non-radial maximizers

Of course this solution is not a ground state of the Rayleigh quotient. The idea of the author is to find it as a critical point of the Rayleigh quotient (1.2) restricted to a wellchosen subspace of H01 invariant under the action of some subgroup of O(N ). In this paper we are interested in the critical case when the spatial dimension N is equal to 2. In this case any polynomial growth is allowed and the critical growth is given by the Trudinger-Moser inequality (see [16], [10]), namely Z 2 sup eβu dx ≤ C(β) ≤ C(4π) < +∞ for β ≤ 4π, (1.4) kukH 1 ≤1 0

B(0,1)

Z

2

eβu dx = +∞

sup kukH 1 ≤1 0

for

β > 4π.

(1.5)

B(0,1)

The second relation (1.5) can be easily established by testing the functional Z 2 eβu dx B(0,1)

on a family of functions called Moser’s sequence defined as follows: p    log k

|x|
0, p > 0 and 1 < γ ≤ 2. We are interested in understanding for which values of γ, p and α the supremum of F (u) on the set S1 = {u ∈ H01 : kukH01 ≤ 1} is finite

340

M. Calanchi, E. Terraneo

and attained. Then, as in the polynomial case, the natural question is to know whether it is achieved by a radial or by a non-radial function. We denote by Tα,p,γ = sup F (u) u∈ S1

and by R Tα,p,γ =

sup

F (u),

u∈ S1 , u rad

namely the supremum considered only over the sub-space of radial functions. Thanks to the Trudinger-Moser inequality, Tα,p,γ is finite in the subcritical cases 1 < γ < 2, p > 0 and γ = 2, 0 < p < 4π, and in the critical case γ = 2 and p = 4π. For these cases we prove R that the supremum Tα,p,γ on radial functions differs from that on non-radial functions, provided that α is large enough. Indeed we establish the theorem: Theorem 1.1 For p > 0, 1 < γ < 2 and for 0 < p ≤ 4π, γ = 2, there exists α∗ > 0 such R for any α > α∗ . that Tα,p,γ > Tα,p,γ By an argument similar to that used in Lemma 2.1 in [6] one can prove that in the subcritical R are cases p > 0, 1 < γ < 2 and 0 < p < 4π, γ = 2, both suprema Tα,p,γ and Tα,p,γ achieved. Then in these cases an analogue of the result of Smets, Su and Willem holds: Corollary 1.1 For p > 0, 1 < γ < 2 and 0 < p < 4π, γ = 2 there exists α∗ > 0 such that no maximizer of F (u) on S1 is radial provided that α > α∗ . Then we analyse the supercritical case, namely γ = 2 and p > 4π, and we show that the supremum over the whole space H01 is not finite. In order to prove this it is enough to evaluate the functional F on a suitable family of Moser’s type functions supported in balls of radius tending to zero and centered at points approaching the boundary. In this way the action of the weight |x|α becomes negligible and we are in the same situation described by the Trudinger-Moser inequality (1.5). Moreover, in analogy with the result due to Ni in [11], we can prove that for 4π < p ≤ 4π + 2πα the supremum over the subspace of radial functions in H01 is finite. In fact we establish the following result: Proposition 1.1 i) Tα,p,2 = +∞ if p > 4π; R R ii) Tα,p,2 < +∞ if 0 < p ≤ 4π + 2πα and Tα,p,2 = +∞

if p > 4π + 2πα.

Finally we propose for γ = 2 a sort of generalization of the result described in Theorem 1.1. We consider the functional: Z 2 Gλ (u) = |x|α (epu − 1 − λpu2 )dx (1.9) B(0,1)

341

Non-radial maximizers

where λ ∈ [0, 1], α > 0 and 0 < p ≤ 4π and denote Tα,λ =

sup kukH1 ≤1 0

Gλ (u) =

sup

Gλ (u)

(1.10)

kukH1 =1 0

(where the last equality is due to the monotonicity properties of the functional Gλ ). We prove the following: Theorem 1.2 Let 0 < p ≤ 4π. Then there exists 0 ≤ λ∗ < 1 such that for any R λ ∈ [λ∗ , 1], Tα,λ > Tα,λ , for large values of α. The presence of the term λpu2 in the functional (1.9) seems to us to be in a certain way necessary: in fact we do not know if λ∗ can be equal to zero. R For the subcritical case we also prove that both suprema Tα,λ and Tα,λ are achieved and so we obtain: Corollary 1.2 For any 0 < p < 4π, there exists 0 ≤ λ∗ < 1 such that for any λ ∈ [λ∗ , 1] no maximizer of Gλ (u) on S1 is radial for large value of α. Finally we point out that any maximizer of (1.10) satisfies for some constant c the following elliptic problem  2   −∆u = c|x|α u(epu − λ) in B(0, 1), u > 0 in B(0, 1),   u| ∂B(0,1) = 0.

(1.11)

Of course the constant c cannot be computed explicitly because of the non-homogeneity of the exponential function. So there is no direct correspondence between equation and ground state solutions (as it happens for the H´enon equation). The question of existence of non-symmetric solutions for symmetric problems has been widely investigated (we refer the reader to the survey of Brezis ([2]) for a complete bibliography). Here we only mention the paper of Brezis and Nirenberg [3], in which the authors prove that if the domain is an annulus there exists a non-radial solution for (1.1) in the autonomous case (i.e α = 0). The results of existence for the H´enon equation in [14] and [13] and in Theorem 1.1 above suggest that the coefficient |x|α has a similar effect to the presence of a ”hole” in B(0, 1), when α is large enough. We also want to remark that in spite of these ”simmetry breaking phenomena” the maximizers may still present a weaker symmetry, as suggested in the paper of Smets and Willem [15].

342

2

M. Calanchi, E. Terraneo

Proof of Theorem 1.1

In the present section we consider the subcritical cases 1 < γ < 2, p > 0 and γ = 2, 0 < p < 4π, and the critical case γ = 2 and p = 4π and we prove that the supremum on R non-radial functions Tα,p,γ is greater than the supremum on radial functions Tα,p,γ at least R for α large enough. Our first result is an asymptotic estimate of Tα,p,γ for α → +∞. Lemma 2.1 (Asymptotic estimate). For p > 0, 1 < γ < 2 and 0 < p ≤ 4π, γ 2 R γ = 2, then Tα,p,γ ∼ 2 S(2γ)p , for α → +∞ where S(2γ) is the Sobolev constant R α1+γ 2γ S(2γ) = supkukH 1 ≤1 B(0,1) |u| dx. 0

The proof of Lemma 2.1 relies on the following lemma. Lemma 2.2 There exists a positive constant C such that for any u ∈ H10 and any q ≥ 2 the following inequality holds: h q i q1 kukLq ≤ C Γ +1 kukH10 , 2 where Γ(λ) =

R +∞ 0

(2.12)

e−x xλ−1 dx is the Gamma function.

Proof of Lemma 2.2. Since H10 ,→ Leu2 −1 , where Leu2 −1 is the Orlicz space defined 2 via the convex function ϕ(t) = et − 1, t ∈ [0, +∞), the inequality (2.12) is a direct consequence of i q1 h q +1 kukL u2 , (2.13) kukLq ≤ Γ e −1 2 established in [12]. For the convenience of the reader we recall in the appendix the definition of the Orlicz space Leu2 −1 and the proof of the inequality (2.13).  Proof of Lemma 2.1. For a radially symmetric function consider the transformation √ 2 , introduced in the paper of Smets, Su and Willem [14]. u(|x|β ) = β w(|x|) , β = α+2 By an easy computation we have Z Z  γ γ    γ γ epβ 2 |w| − 1 − pβ 2 |w|γ dx; |x|α ep|u| − 1 − p|u|γ dx = β B(0,1)

B(0,1)

and

Z

|∇w(x)|2 dx ≤ 1.

B(0,1)

So we obtain R Tα,p,γ =

Z sup kwkH1 ≤1, w rad 0

β B(0,1)



 γ γ γ epβ 2 |w| − 1 − pβ 2 |w|γ dx.

(2.14)

343

Non-radial maximizers

The control from below of the integral in (2.14) comes from the following inequality: R Tα,p,γ ≥

β 1+γ p2 2 kwkH 1 ≤1

Z

|w|2γ dx =

sup 0

B(0,1)

β 1+γ p2 S(2γ). 2

R In order to obtain an estimate from above of Tα,p,γ we expand the exponential function in series to get:

Z



β

+∞ Z  γ X γ γ epβ 2 |w| − 1 − pβ 2 |w|γ dx = β

B(0,1)

=

k=2

β 1+γ p2 2

Z

|w|2γ dx + β

B(0,1)

+∞ Z X k=3

X β 1+γ p2 S(2γ) + β 2

k=3

B(0,1)

B(0,1)

k

k

γ

B(0,1)

+∞ Z

≤

γ

(pβ 2 |w|γ ) dx k!

(pβ 2 |w|γ ) dx k!

γ

k

(pβ 2 |w|γ ) dx. k!

Now we estimate the series: thanks to Lemma 2.2 we have γk kukγk γk ≤ C Γ(

γk + 1)kukγk H10 2

so in order to control the integral in (2.14) from above it is enough to show that the series   γ k +∞ X (pβ 2 ) γk Γ + 1 C γk k! 2

k=3

converges, since kukH10 ≤ 1. We recall that the Gamma function has a minimum localized between 1 and 2 and it is increasing afterwards. So the last series is controlled by γ +∞ X pβ 2

k=3


k

C γk Γ(k + 1) 3γ = O(β 2 ) k!

(2.15)

for β small enough. So we obtain the estimate from above R Tα,p,γ ≤

3γ β 1+γ p2 S(2γ) + O(β 1+ 2 ). 2

 Now we are able to prove Theorem 1.1. Proof of Theorem 1.1. Let u be a positive smooth function with support in B(0, 1) and such that kukH10 = 1. Following Smets, Su and Willem let us consider uα (x) =

344

M. Calanchi, E. Terraneo

u(α(x − xα )), where xα = (1 − α1 , 0). Since kuα kH10 = 1 and it has support in the ball B(xα , α1 ), by the change of variables y = α(x − xα ) we obtain: Z   γ Tα,p,γ ≥ |x|α ep|uα | − 1 − p|uα |γ dx B(0,1) Z   γ |x|α ep|uα | − 1 − p|uα |γ dx = 1 ) B(xα , α

 ≥

2 α

1−

α

1 α2

Z



γ

ep|u| − 1 − p|u|γ



dy

B(0,1) 2γ S(2γ)p2 , α1+γ

R ∼ and so Tα,p,γ ≥ αC2 , for α → +∞. Since, by Lemma 2.1, Tα,p,γ R we are able to conclude that Tα,p,γ > Tα,p,γ for α large enough.

as α → +∞ 

Proof of Corollary 1.1. In view of Theorem 1.1 it is enough to show that both suprema R Tα,p,γ and Tα,p,γ are achieved. For simplicity we only consider the case of the supremum Tα,p,γ for γ = 2, 0 < p < 4π. Let {un } be a maximizing sequence such that kun kH01 ≤ 1. Then, up to a subsequence, there exists u ∈ H01 such that un * u weakly in H01 and strongly in Lq , ∀q ≥ 1 and un (x) → u(x) a.e. in B(0, 1). By using Lemma 2.1 in [6] in order to prove that Z Z  2   2  α pun 2 |x| e − 1 − pun dx → |x|α epu − 1 − pu2 dx as n → +∞ B(0,1)

B(0,1)

it is enough to show that Z

2

|x|α epun |un |dx ≤ C.

(2.16)

B(0,1)

Since p < 4π, there exists q > 1 such that pq < 4π. By the H¨older inequality, if q 0 is the conjugate exponent of q, one has Z

α pu2n

|x| e B(0,1)

! 10 q

Z |un |dx ≤

|un | B(0,1)

q0

! q1

Z

pqu2n

e

≤C

B(0,1)

which is a direct a consequence of the Trudinger-Moser inequality.



Remark. We do not know if Corollary 1.1 holds in the case p = 4π and γ = 2. In fact ´ and Ruf below, it is thanks to equality (2.14) and to the Theorem of de Figueiredo, do O R easy to show that the supremum Tα,4π,2 is achieved. On the other hand it is not clear if the supremum Tα,4π,2 is attained or not. ´ Ruf [5]] Suppose that g satisfies: Theorem [de Figueiredo, do O, 1 i) g ∈ C (R); ii) g is increasing on R+ and g(s) = g(|s|);

345

Non-radial maximizers

2

iii) 0 ≤ g(s) ≤ e4πs − 1, for any s ≥ 0; iv) g(s) has subcritical growth. Then Z sup kukH 1 ≤1 0

g(u)dx

B(0,1)

is attained.

3

A remark on the supercritical case

In the present section we prove that in the supercritical case γ = 2, p > 4π the supremum Tα,p,2 is not finite. In order to prove this we evaluate the functional F (u) on some particular functions obtained by a suitable translation and dilation of Moser’s functions in a region of B(0, 1) far from the origin where the presence of |x|α can be neglected. Proof of Proposition 1.1 (Case i). Consider the following family of functions p  log k      1 ) 1  log( α|x−x α| wk,α (x) = √ √  2π  log k     0

|x − xα |
2, k > 2. We observe that kwk,α kH01 = 1. Moreover, Z

 2  2 |x|α epwk,α − 1 − pwk,α dx

Tα,p,2 ≥ B(0,1)

Z

  2 2 |x|α epwk,α − 1 − pwk,α dx

≥

1 ) B(xα , kα α

 ≥

1− 

=

1−

2 α

Z

2 α

α 



p

e 2π log k − 1 −

1 B(xα , kα ) p

k 2π − 1 −

 p log k dx 2π

 π p log k 2π α2 k 2

and the last term tends to +∞ when k → +∞ since p > 4π.



If we restrict our attention to the supremum on radial functions the situation is different. In this case the factor |x|α , α > 0 improves the convergence properties of the integral and we have the following result: Proof of Proposition 1.1 (Case ii). For a radially symmetric function, using the same

346

M. Calanchi, E. Terraneo

change of variables as in Proposition 2: u(|x|β ) = R Tα,p,2 =

Z sup



β

||w||H 1 ≤1, w rad

√

βw(|x|) , β =

2 α+2 ,

we get

 2 epβw − 1 − pβw2 dx.

(3.18)

B(0,1)

0

Thanks to the Trudinger-Moser inequality quoted above this quantity is finite if and only if p verifies 0 < p ≤ 4π  β = 2πα + 4π. R Remark. We point out that the supremum on radial functions Tα,p,2 is attained for any 0 < p < 4π + 2πα, α > 0. Indeed, this result is a direct consequence of equality (3.18) ´ Ruf [5] quoted at the end of the second section. and of the Theorem of de Figueiredo, do O, In the limit case p = 4π + 2πα, α > 0 it is not known if the supremum is achieved (see Theorem 5 in [5]).

4

A generalization

In the present section we consider Z Tα,λ =

 2  |x|α epu − 1 − pλu2 dx

sup kukH1 ≤1 0

(4.19)

B(0,1)

where α > 0, 0 < p ≤ 4π and 0 ≤ λ ≤ 1 and we prove Theorem 1.2, which generalizes Theorem 1.1 for γ = 2. Proof of Theorem 1.2. The case λ = 1 was treated in Theorem 1.1; so here we only analyse the case λ < 1. At first, in a similar way as in Lemma 2.1, we establish the following asymptotic estimate for the supremum on radial functions in the unit ball of H01 : R Tα,λ ∼

4p(1 − λ) α 2 λ1

as α → +∞,

where λ1 is the first eigenvalue of the Laplacian in H01 (B(0, 1)). Indeed, thanks to the √ 2 , we obtain transformation u(|x|β ) = β w(|x|), β = α+2 R Ta,λ

Z =

sup kwkH 1 ≤1,w rad 0



β

 2 epβw − 1 − pβλw2 dx.

B(0,1)

Now the estimate from below comes from the inequality: R Ta,λ ≥

sup kwkH 1 ≤1 0

β 2 p(1 − λ)

Z B(0,1)

w2 dx =

pβ 2 (1 − λ) . λ1

(4.20)

347

Non-radial maximizers

On the other hand we control the supremum from above by using a Taylor expansion of the exponential term, namely Z   2 β epβw − 1 − pβλw2 dx = B(0,1)

Z

2

2

= β p(1 − λ)

w dx + β B(0,1)

+∞ k k Z X p β k=2

k!

(4.21) w

2k

dx.

B(0,1)

Then, thanks to Lemma 2.2, there exists a constant C > 0 independent of w such that 2k 2k kwk2k 2k ≤ C Γ(k + 1)kwkH 1 , 0

so, since kwkH01 ≤ 1, we get +∞

X (pβC 2 )k β 2 p(1 − λ) +β Γ(k + 1) λ1 k!

R Tα,λ ≤

k=2

(4.22)

β 2 p(1 − λ) p2 C 4 = + β3 λ1 1 − pβC 2 2 for β = α+2 small enough. For the supremum on the whole unit ball in H01 , since λ ≤ 1, one has

Z Tα,λ =

 2  |x|α epu − 1 − pλu2 dx

sup kukH1 ≤1 0

B(0,1)

Z ≥

sup kukH1 ≤1 0

 2  |x|α epu − 1 − pu2 dx

B(0,1)

and as in the proof of Theorem 1.1 we can show that there exists C > 0 independent of λ such that Tα,λ ≥ αC2 , for α large enough. The conclusion of the theorem follows by ∗ ) choosing λ∗ such that 4p(1−λ < C.  λ1 Proof of Corollary 1.2. The proof follows the same lines of Corollary 1.1.



As mentioned in the introduction we are not able to establish Theorem 1.2 if λ = 0. Indeed in this case we obtain that R Tα,0 ∼

4p , as α → +∞; α 2 λ1

on the other hand for the supremum on all functions Tα,0 we give an estimate from below, namely Tα,0 ≥ αC2 for α big enough with a constant C < λ4p1 . So the same argument as in Theorem 1.2 cannot apply.

348

5

M. Calanchi, E. Terraneo

Appendix

In this last section we recall the definition of the Orlicz space Leu2 −1 and we give the proof of equality (2.13) established in [12]. We refer to [1] and [9] for a wide presentation of the Orlicz spaces. Let ϕ : [0, +∞[→ [0, +∞[ be a convex, strictly increasing function such that lim+ ϕ(s) = s→0

ϕ(0) = 0 and lim ϕ(s) = +∞. s→+∞

Definition 5.1 The Orlicz space Lϕ is defined by Lϕ = {u measurable on B(0, 1) such that ∃ K > 0 )   u(x) dx < +∞ . with ϕ K B(0,1) Z

It can be proven that Lϕ is a vector space (see [9]). For every u ∈ Lϕ , let us define the functional ( )   Z u(x) kukϕ = inf K > 0 such that ϕ dx ≤ 1 K B(0,1) One can prove that k · kϕ is a norm on Lϕ and (Lϕ , k · kϕ ) is a Banach space (see [9]). It is easy to verify that the function ϕ(u) = |u|p , 1 < p < +∞ defines the Lp space. 2 In the following we will consider the Orlicz space defined by ϕ(u) = eu − 1. For this particular space we have: Lemma 5.1 For every p ≥ 2 we have Leu2 −1 ,→ Lp and i p1 h p +1 kukL 2 . kukLp ≤ Γ eu −1 2 where Γ(λ) =

R +∞ 0

(5.23)

e−x xλ−1 dx.

Proof. Let K = kukL

2 eu −1

> 0, otherwise inequality (5.23) is obvious. We will use the

property that ex − 1 ≥

xρ , Γ(ρ + 1)

(5.24)

for every ρ ≥ 1 and x ≥ 0. This is easily proven when ρ is an integer. For x ∈ (0, 1] and for every ρ > 1 the inequality follows by using the fact that Γ(ρ + 1) ≥ 1 if ρ ≥ 1, and ex − 1 ≥ xρ . For x ∈ [1, +∞) and 1 < ρ < 2 the inequality (5.24) is implied by the fact that ex − 1 − x2 ≥ 0 and the property Γ(ρ + 1) ≥ 1. The general inequality for x ∈ [1, +∞) and 2 < ρ follows from the case 1 < ρ < 2 by an easy computation, since

349

Non-radial maximizers

Γ(ρ + 1) = ρΓ(ρ). By the theorem of monotone convergence for K = kukL

2 eu −1

have

Z



1≥

> 0 we

 |u| 2 e( K ) − 1 dx

B(0,1)

and by (5.24)  Z 1≥ B(0,1)

for every ρ ≥ 1. Thus we obtain 1 ≥

|u| K

2ρ

Γ(ρ + 1)

kuk2ρ 2ρ 2ρ K Γ(ρ+1)

dx

and we conclude that K ≥

kuk2ρ 1

(Γ(ρ+1)) 2ρ

.

Acknowledgments. The authors would like to thank Bernhard Ruf and Enrico Serra for bringing this problem to their attention and for stimulating discussions.

References [1] R.A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [2] H. Brezis, Symmetry in Nonlinear PDE’s, Differential Equations: La Pietra 1996 (Florence), Volume 65 of Proc. Sympos. Pure Math., pages 1–12, Amer. Math. Soc., Providence, RI, 1999. [3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (4) (1983); 437–477. [4] L. Carleson and S.A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (2) (1986); 113–127. ´ and B. Ruf, On an inequality by N. Trudinger and [5] D.G. de Figueiredo, J.M. do O, J. Moser and related elliptic equations, Comm. Pure Appl. Math. 55 (2) (2002); 135–152. [6] D.G. de Figueiredo, O.H. Miyagaki, and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (2) (1995); 139–153. [7] M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (3) (1992); 471–497. [8] B. Gidas, W.M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (3) (1979); 209–243.

350

M. Calanchi, E. Terraneo

[9] L. Maligranda, Orlicz Spaces and Interpolation. IMECC, 1989. [10] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71); 1077–1092. [11] W.M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J. 31 (6) (1982); 801–807. [12] B. Ruf and E. Terraneo, The Cauchy Problem For a Semilinear Heat Equation With Singular Initial Data, Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Volume 50, Progr. Nonlinear Differential Equations Appl., pages 295– 309. Birkh¨auser, Basel, 2002. [13] E. Serra, Non radial positive solutions for the H´enon equation with the critical growth, To appear in Calc. Var. PDEs. [14] D. Smets, J. Su, and M. Willem, Non-radial ground states for the H´enon equation, Commun. Contemp. Math. 4 (3) (2002); 467–480. [15] D. Smets, and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. PDEs 18 (2003); 57–75. [16] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967); 473–483.