Multiple solutions for operator equations involving duality map

Dedicated to Professor Philippe G. Ciarlet on his 70th birthday

MULTIPLE SOLUTIONS TO OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS ON ORLICZ-SOBOLEV SPACES VIA THE MOUNTAIN PASS THEOREM GEORGE DINC€ and PAVEL MATEI

Let

X

be a real reexive smooth Banach space having the Kade£-Klee property,

compactly imbedded in a real Banach space functional. By using the

V

and let

G:V →R

a dierentiable

of the Mountain Pass Theorem (Rabinowitz 0 [17]), the multiplicity of solutions to operator equation Jϕ u = G (u), where Jϕ is the duality mapping on

Z2 -version

X , corresponding to the gauge function, ϕ is studied.

Sim-

ilar results are obtained in [12] and [13] by using the fountain theorem (Bartsch [3]) and the dual fountain theorem (Bartsch-Willem [4]), respectively. Equations of the above form with

Jϕ

a duality mapping on Orlicz-Sobolev spaces, are con-

sidered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces may be derived.

AMS 2000 Subject Classication: 35B38, 47J30. Key words: critical point, mountain pass theorem, duality mapping, OrliczSobolev space.

1.

INTRODUCTION

This paper is concerned with multiplicity results for equations of the type (1.1)

Jϕ u = G0 (u),

where (i) X is a real reexive and smooth Banach space having the Kade£-Klee property, compactly imbedded in the real Banach space V ; (ii) Jϕ : X → X ∗ is a duality mapping corresponding to the gauge function ϕ (see Denition 1, below); (iii) G0 : V → V ∗ is the dierential of the functional G : V → R. As usual, X ∗ (resp. V ∗ ) denotes the dual space of X (resp. V ) and h· , ·iX,X ∗ (resp. h· , ·iV,V ∗ ) denotes the duality pairing between X ∗ and X REV. ROUMAINE MATH. PURES APPL.,

53 (2008), 56, 419437

420

George Dinc  and Pavel Matei

2

(resp. V ∗ and V ). Often, we shall omit to indicate the spaces in duality and, simply, we shall write h· , ·i. Our approach is a variational one, the Z2 -version of the Mountain Pass Theorem due to Rabinowitz ([17]) being the basic ingredient which is used. Equations of the form (1.1) with Jϕ a duality mapping on Orlicz-Sobolev spaces, are considered as applications. As particular cases of these results, some multiplicity results concerning duality mappings on Sobolev spaces are derived. Moreover, these results apply to many dierential operators which in fact are duality mappings on some appropriate spaces of functions (for example, if ∆p , 1 < p < ∞, is the so called p-Laplacian, then −∆p is the duality mapping on W01,p (Ω) corresponding to the gauge function ϕ(t) = tp−1 , t ≥ 0). 2.

THE MAIN RESULT

Theorem 1. Let X be a real reexive smooth Banach space having the Kade£-Klee property and compactly imbedded in a real Banach space V. Let H ∈ C 1 (X, R) be an even functional of the form

H = Ψ − G,

(2.1)

where

(i) Ψ(u) = Φ(kuk) at any u ∈ X with Z t (2.2) Φ(t) = ϕ(ξ) dξ ,

∀t ≥ 0,

0

ϕ : R+ → R+ being a gauge function which satises p∗ = sup tϕ(t) Φ(t) < ∞; t>0

(ii) G : V → R satises: (ii)0 G(0) = 0; (ii)1 G0 : V → V ∗ is demicontinuous; (ii)2 there is a constant θ > p∗ such that, at any y ∈ V ,

0  (2.3) G (y), y V,V ∗ − θG(y) ≥ C = const.; (iii) there exists c0 > 0 such that for any u ∈ X with kukX < c0 one has (2.4)

H(u) > c1 kukpX − c2 ki(u)kqV ,

where i stands for the compact injection of X in V while 0 < p < q and c1 > 0, c2 > 0; (iv) for any nite dimensional subspace X1 ⊂ X , there exist real constants d0 > 0, d1 , d2 > 0, d3 , s > 0 and r < s (generally depending on X1 ) such that (2.5)

H(u) ≤ d1 kukrX − d2 kuksX + d3 ,

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Equations involving duality mappings on Orlicz-Sobolev spaces

421

for any u ∈ X1 with kukX > d0 . Then the functional H possesses an unbounded sequence of critical values. Before proceeding to the proof of Theorem 1, we list some results we need. First, we recall that a real Banach space X is said to be smooth if it has the following property: for any x ∈ X , x 6= 0, there exists a unique u∗ (x) ∈ X ∗ such that hu∗ (x), xi = kxkX and ku∗ (x)kX ∗ = 1. It is well known (see, for instance, Diestel [8], Zeidler [22] ) that the smoothness of X is equivalent to the Gâteaux dierentiability of the norm. Consequently, if (X, k · kX ) is smooth, then for any x ∈ X , x 6= 0, the only element u∗ (x) ∈ X ∗ with the properties hu∗ (x), xi = kxkX and ku∗ (x)kX ∗ = 1 is u∗ (x) = k · k0X (x) (where k · k0X (x) denotes the Gâteaux gradient of the k · kX -norm at x). A function ϕ : R+ → R+ is said to be a gauge function if ϕ is continuous, strictly increasing, ϕ(0) = 0, and ϕ(t) → ∞ as t → ∞.

Denition 1. If X is a real smooth Banach space and ϕ : R+ → R+ is a gauge function, the duality mapping on X corresponding to ϕ is the mapping Jϕ : X → X ∗ dened by (2.6)

Jϕ 0 = 0, Jϕ x = ϕ (kxkX ) k · k0X (x) if x 6= 0. The following metric properties are consequences of Denition 1:

(2.7)

kJϕ xkX ∗ = ϕ (kxkX ) ,

hJϕ x, xi = ϕ (kxkX ) kxkX ,

∀x ∈ X.

For the main properties of duality mappings, see [6], [9], [22]. In order to state the next result, we recall that if X is a real Banach space and H ∈ C 1 (X, R), we say that H satises the Palais-Smale condition on X ((P S)-condition, for short) if any sequence (un ) ⊂ X with (H(un )) bounded and H 0 (un ) → 0 as n → ∞, possesses a convergent subsequence. The basic result we need for proving Theorem 1 is the Z2 -version of the Mountain Pass Theorem due to Rabinowitz: Theorem 2 (Rabinowitz [17, Theorem 9.12]). Let X be an innite dimensional real Banach space. Assume H ∈ C 1 (X, R) is even, satises the (P S)-condition and H(0) = 0. If (G1 ) there exist ρ > 0 and r > 0 such that H(u) ≥ r for kuk = ρ; (G2 ) for each nite dimensional subspace X1 of X the set {u ∈ X1 | H(u) ≥ 0} is bounded, then H possesses an unbounded sequence of critical values.

Now, we are able to give the proof of Theorem 1. The idea is that the hypotheses of Theorem 1 entail those of Theorem 2. Indeed, the following result holds (see [12, Corollary 2]):

422

George Dinc  and Pavel Matei

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Proposition 1. Let X be a real reexive smooth Banach space having the Kade£-Klee property and compactly imbedded in a real Banach space V . Let H ∈ C 1 (X, R) be a functional of the form

(2.8)

H = Ψ − G,

where

(i) Ψ(u) = Φ(kuk) at any u ∈ X with Z t ϕ(s) ds, Φ(t) =

∀t ≥ 0,

0

and ϕ : R+ → R+ being a gauge function which satises sup t>0

tϕ(t) = p∗ < ∞; Φ(t)

(ii) G : V → R satises: (ii)1 G0 : V → V ∗ is demicontinuous; (ii)2 there is a constant θ > p∗ such that

0  (2.9) G (y), y Y,Y ∗ − θG(y) ≥ C = const.

∀y ∈ V .

Then H satises the (P S)-condition. Proof. It is now clear that, under the hypotheses of Theorem 1, the fact that H satises the (P S)-condition is a direct consequence of Proposition 1. We will show that hypothesis (G)1 of Theorem 2 is satised. Indeed, taking into account that ki(u)kV ≤ ckukX , ∀x ∈ X , it follows from (2.4) that   H(u) > kukpX · c1 − c2 cq kukq−p X for all u ∈ X with kukX < c0 . So, for

 kukX = ρ ≤ min c0 ,

c1 2cq c2



1 q−p

we have

! ,

C p ρ > 0, 2 that is, hypothesis (G1 ) of Theorem 2 is fullled with r = C2 ρp . On the other hand, let X1 be a nite dimensional subspace of X . We shall show that the set S = {u ∈ X1 | H(u) ≥ 0} is bounded. Indeed, taking into account (2.5), if u ∈ S , kukX > d0 , then H(u) >

(2.10)

d1 kukrX − d2 kuksX + d3 ≥ 0.

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Equations involving duality mappings on Orlicz-Sobolev spaces

423

Since s > r, we conclude that S is bounded, thus hypothesis (G2 ) of Theorem 2 is fullled. The conclusion follows from Theorem 2.  3.

APPLICATIONS TO ORLICZ-SOBOLEV SPACES

Throughout this section Ω denotes a bounded open subset of RN , N ≥ 2. Let a : R→ R be a strictly increasing odd continuous function with lim a(t) = t→+∞

+∞. For m ∈ N∗ let us denote by W0m EA (Ω) the Orlicz-Sobolev space generated by the N -function A, dened by Z t a(s) ds. (3.1) A(t) = 0

We shall always suppose that

Z

(3.2)

1

A−1 (τ )

dτ < ∞, N +1 τ N replacing, if necessary, A by another N -function equivalent to A near innity (which determines the same Orlicz space). Suppose also that Z t −1 A (τ ) (3.3) lim dτ = ∞. t→∞ 1 τ NN+1 With (3.3) satised, we dene the Sobolev conjugate A∗ of A by setting Z t −1 A (τ ) −1 (3.4) A∗ (t) = dτ , t ≥ 0. N +1 0 τ N The existence and multiplicity of weak solutions to the boundary value problem X (3.5) Ja u = (−1)|α| Dα gα (x, Dα u) in Ω,

lim

t→0 t

|α| 0 and θα > p∗ = sup ta(t) A(t) t>0

such that (3.9)

0 < θα Gα (x, s) ≤ sgα (x, s)

for a.e. x ∈ Ω and all s with |s| ≥ sα , where Z s (3.10) Gα (x, s) = gα (x, τ ) dτ . 0

Assume also that (H)3 the function a(t) t is nondecreasing on (0, ∞), (3.2) and (3.3) being fullled as well (see the beginning of this section); By (weak) solution to problem (3.5), (3.6) we understand a solution to the equation

Ja u = G0 (u),

(3.11)

in the following functional framework: T (i) X = W0m EA (Ω) normed with k·km,A , V = W m−1 LMβ (Ω) normed |β| 1. Thus, Mα , |α| < m, increase essentially more slowly than A∗ . Hypothesis (H)2 is covered by (iv) (with gα odd functions in the second argument, according to (ii). In order to prove that A and A satisfy the ∆2 -condition, the result below is needed (see [11, Lemma 8.1, (i)]).

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Equations involving duality mappings on Orlicz-Sobolev spaces

433

R |t| 3. Let A : R → R+ , A(t) = 0 a(s) ds, be an N -function and A the complementary N -function to A. Assume that Lemma

p∗ = sup t>0

ta(t) 0

ta(t) > 1. A(t)

Then both A and A satisfy the ∆2 -condition. In our case, as one has already seen, p∗ = pn < N and p0 = p1 > 1 (according to (i)). Since M α (s) =

0

|s|qα 1 0 , q qα α

+ q10 = 1, |α| < m, s ∈ R, it is easy α

to check (by denition) that M α , |α| < m, satisfy the ∆2 -condition. On the other hand, since p0 = p1 , (4.6) says that (H)4 in Theorem 3 is satised. pn Finally, since p0 = p1 < pn < NN−p = p∗ , (H)6 is satised too. n The result now follows by Theorem 3.  ses:

Example 2. Consider problem (3.5), (3.6), under the following hypothe-

√ (i) the function a : R → R is dened by a(t) = |t|p−2 t t2 + 1, 2 ≤ p < N − 1; (ii) the Carathéodory functions gα : Ω × R → R, |α| < m, are odd in the second argument, that is, gα (x, −s) = −gα (x, s), and satisfy the conditions lim sup s→0

λ1,p gα (x, s) < , a (s) 2pN0

|α| < m,

uniformly with respect to almost all x ∈ Ω, where λ1,p are given by (3.13) and P 1; N0 = |α| 0 and θα > p + 1 such that conditions (4.3) hold. Under these assumptions, problem (3.5), (3.6) has a sequence of weak solutions.

Proof. The idea of the proof is that used in Example 1, namely, we shall

show that the assumptions made entail those of Theorem 3. √ p−2 t2 + 1 First, we prove that hypothesis (H)3 is satised. Since a(t) t =t for all t > 0, the function a(t) t is nondecreasing on (0, ∞). In order to prove that (3.2) and (3.3) are satised, we shall use Lemma 2. In our case, p∗ = p + 1 ([11, Example 8.6]) and p + 1 < N (by (i)). Since a(t) ≥ tp−1 , t > 0, one has 1 (4.8) A(t) ≥ tp , ∀t > 0. p

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George Dinc  and Pavel Matei

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Therefore, (4.4) is satised with C = p1 , γ = p < N and any δ > 0. So, A∗ exists and we can compute p∗ . We obtain

p∗ = lim inf t→∞

tA0∗ (t) N (p + 1) = . A∗ (t) N −p−1

Second, the hypothesis (H)1 in Theorem 3 is satised with Mα (s) = |α| < m, which, obviously, satisfy the ∆2 -condition. Also, Mα , |α| < m, increase essentially more slowly than A∗ near innity. Indeed, as in (4.7), |s|qα qα ,

A∗−1 (t) s = lim cα . 1 −1 +1 s→∞ t→∞ Mα (t) (A(s)) qα N lim

It is sucient to show that

s

lim

s→∞

1

(A(s)) qα

Since a(t) ≥ tp , ∀t ≥ 0, we have A(t) ≥

lim

s→∞

s (A(s))

1 1 +N qα

1 +N

tp+1 p+1 ,

∀t ≥ 0. Consequently, s

≤ lim

s→∞

= 0.

(p + 1)

1 1 +N qα



·s 

1 1 +N qα

 (p+1)

= 0,

 as by (iii), the degree of denominator is (p + 1) q1α + N1 > 1. The hypothesis (H)2 is covered by (iv) (with gα odd functions in the second argument, according to (ii). In order to prove that A and A satisfy the ∆2 -condition, we shall use Lemma 3. In our case, as one has already seen, p∗ = p + 1 < N and p0 = p > 1 0 qα

(according to (i)). Also, the functions M α (s) = |s|q0 , q1α + q10 = 1, |α| < m, α α s ∈ R, satisfy the ∆2 -condition. On the other hand, since p0 = p (see [11, Example 8.6]), (4.8) says that (H)4 in Theorem 3 is satised. (p+1) Finally, since p0 = p < p + 1 < N N −p−1 = p∗ , (H)5 is satised, too. The result now follows by Theorem 3.  ses:

Example 3. Consider problem (3.5), (3.6), under the following hypothe-

(i) the function a : R → R is dened by a(t) = |t|p−2 t ln (1 + α + |t|), 2 ≤ p ≤ N − 1, α > 0; (ii) the Carathéodory functions gα : Ω × R → R, |α| < m, are odd in the second argument, that is, gα (x, −s) = −gα (x, s), and satisfy the conditions:

lim sup s→0

ln(1 + α)λ1,p gα (x, s) < , a (s) 2pN0

|α| < m,

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435

uniformly with respect to almost all x ∈ Ω, where λ1,p are given by (3.13) and P N0 = 1; |α| 0 and θα > p + 1 such that conditions (4.3) hold. Under these asumptions, problem (3.5), (3.6) has a sequence of weak solutions.

Proof. The idea of the proof is that used in Example 1, namely, we shall show that the assumptions made entail those of Theorem 3. p−2 ln(1 + First, we prove that hypothesis (H)3 is satised. Since a(t) t =t a(t) α + t) for all t > 0, the function t is nondecreasing on (0, ∞). In order to prove that (3.2) and (3.3) are satised, we shall use Lemma 2. In our case, p∗ = p + C0 < p + 1 ([11, Example 8.10]) and p + 1 ≤ N (by (i)). Since (see [11, Example 8.10, inequality (8.23)]) (4.9)

A(t) ≥

ln (1 + α) p t , ∀t ≥ 0, p

it follows that (4.4) is satised with C = ln(1+α) , γ = p < N and any δ > 0. p Therefore, A∗ exists and we can compute p∗ . We obtain

p∗ = lim inf t→∞

tA0∗ (t) Np = . A∗ (t) N −p qα

Second, hypothesis (H)1 in Theorem 3 is satised with Mα (s) = |s|qα , |α| < m, which, obviously, satisfy the ∆2 -condition. Also, Mα , |α| < m, increase essentially more slowly than A∗ near innity. As in the preceding two examples, this amounts to show that s lim = 0. 1 +1 s→∞ (A(s)) qα N This last equality is true since (4.9) holds. Therefore, s s lim 1 1 ≤ lim   1 + 1  1 1  = 0, + s→∞ s→∞ + p ln(1+α) qα N (A(s)) qα N · s qα N p

  since, by (iii), the degree of denominator is p q1α + N1 > 1. The necessary arguments in order to prove that hypothesis (H)2 of Theorem 3 is satised are those used in the preceding two examples. In order to prove that A and A satisfy the ∆2 -condition, we shall use Lemma 3. In our case, as one has already seen, p∗ = p + C0 < N and

436

George Dinc  and Pavel Matei

18 0 qα

p0 = p > 1 (according to (i)). Also, the functions M α (s) = |s|q0 , q1α + q10 = 1, α α |α| < m, s ∈ R, satisfy the ∆2 -condition. On the other hand, since p0 = p (see [11, Example 8.6]), (4.9) says that (H)4 in Theorem 3 is satised. p Finally, since p0 = p < NN−p = p∗ , (H)5 is satised, too. The result now follows by Theorem 3. 

REFERENCES

[1]

R.A. Adams, Sobolev Spaces. Academic Press, New YorkSan FranciscoLondon, 1975.

[2]

E. Asplund, Positivity of duality mappings. Bull. Amer. Math. Soc.

[3]

T. Bartsch, Innetely many solutions of a symmetric Dirichlet problem. Nonlinear Anal.

[4]

20 (1993), 12051216.

T. Bartsch, M. Willem, On an elliptic equation with concave and convex nonlinearities. Proc. Amer. Math. Soc.

[5] [6]

73 (1967), 200203.

123 (1995), 35553561.

F.E. Browder, Problèmes non-linéaires. Presses de l'Université de Montréal, 1964. Ioana Cior nescu, Duality mappings in nonlinear functional analysis. Publishing House of Romanian Academy, Bucharest, 1974. (Romanian)

[7]

K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, 1985.

[8]

J. Diestel, Geometry of Banach Spaces-Selected Topics. Lecture Notes in Math.

485.

Springer-Verlag, 1975. [9]

G. Dinc , Variational methods and applications, Technical Publ. House, Bucharest, 1980. (Romanian)

[10] G. Dinc  and P. Jebelean, Some existence results for a class of nonlinear equations

involving a duality mapping. Nonlinear Anal.

46 (2001), 347363.

[11] G. Dinc  and P. Matei, Variational and topological methods for operator equations in-

volving duality mappings on Orlicz-Sobolev spaces. Electron J. Dierential Equations

93

(2007), 147. [12] G. Dinc  and P. Matei, Multiple solutions for operator equations involving duality map-

pings on Orlicz-Sobolev spaces. Dierential Integral Equations

21 (2008), 910, 891916.

[13] G. Dinc  and P. Matei, Innitely many solutions for operator equations involving duality

mappings on Orlicz-Sobolev spaces. TMNA (accepted) [14] P.S. Ilias, Nonlinear operator equations in locally convex spaces. Ph.D. Thesis, Univ. Bucharest, 2005. (Romanian) [15] M.A. Krasnosel'skij and Ja.B. Rutitskij, Convex Functions and Orlicz Spaces. Gröningen, Noordho, 1961. [16] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. SpringerVerlag, New York, 1989. [17] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Dif-

ferential Equations. Conference Board of the Math. Sci. Regional Conference Series in Math, no. 65, Amer. Math. Soc., Providence, RI, 1986. [18] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces. Marcel Dekker, New YorkBasel Hong Kong, 1991. [19] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Dierential

Equations. Amer. Math. Soc., Providence RI, 1997.

19

Equations involving duality mappings on Orlicz-Sobolev spaces

437

[20] M. Tienari, A degree theory for a class of mappings of monotone type in Orlicz-Sobolev

spaces. Ann. Acad. Sci. Fenn. Ser. A1 Math. Dissertationes

97 (1994), 168.

[21] M. Willem, Minimax Theorems. Birkhäuser, Basel, 1996. [22] E. Zeidler, Nonlinear Functional Analysis and its Applications II/B: Nonlinear Mono-

tone Operators. Springer, New York, 1990. Received 19 May 2008

University of Bucharest Faculty of Mathematics and Computer Science Str. Academiei 14 010014 Bucharest, Romania Romanian Academy Gheorghe MihocCaius Iacob Institute of Mathematics Statistics and Applied Mathematics Calea 13 Septembrie nr. 13 Bucharest, Romania [email protected] and

Technical University of Civil Engineering Department of Mathematics Bd. Lacul Tei nr. 124 020396 Bucharest, Romania [email protected]