S0362-546X(08)00003-5 10.1016/j.na.2007.12.029 NA 6435

To appear in:

Nonlinear Analysis

Received date: 14 November 2007 Accepted date: 31 December 2007 Please cite this article as: B.E. Breckner, A. Horv´ath, C. Varga, A multiplicity result for a special class of gradient-type systems with non-differentiable term, Nonlinear Analysis (2008), doi:10.1016/j.na.2007.12.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A multiplicity result for a special class of gradient-type systems with non-differentiable term ⋆

of Mathematics and Computer Science, Babe¸s-Bolyai University, Str. M. Kog˘ alniceanu nr. 1, 400084 Cluj-Napoca, Romania

b Department c Faculty

of Mathematics and Computer Science, Petru Maior University, Str. N. Iorga 1, 540088 Tˆ argu Mure¸s, Romania

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a Faculty

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Brigitte E. Breckner a, Alexandru Horv´ath b, Csaba Varga c

of Mathematics and Computer Science, Babe¸s-Bolyai University, Str. M. Kog˘ alniceanu nr. 1, 400084 Cluj-Napoca, Romania

Abstract

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We prove the existence of multiple solutions of certain systems of hemivariational inequalities, using a recent minimax result of B. Ricceri. We apply both our main theorem and the principle of symmetric criticality to obtain multiple solutions of systems of hemivariational inequalities defined on certain Sobolev spaces.

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Key words: locally Lipschitz function, generalized directional derivative, systems of hemivariational inequalities, principle of symmetric criticality, Sobolev spaces 1991 MSC: 47J20, 49J52

Introduction

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Elliptic equations and systems involving p-Laplacians occur in different fields of Mathematical Physics. These problems have been studied intensively in the last years, see, for example, Bartsch and de Figueiredo [1], Bartsch and Wang [2], Boccardo and de Figueiredo [3], Cammaroto, Chinn`ı, and Di Bella [5], ⋆ The research for this paper was supported by the grants CEEX/M3-130/2006 and PN-II-ID-527. Email addresses: [email protected], [email protected] (Brigitte E. Breckner), [email protected] (Alexandru Horv´ ath), [email protected] (Csaba Varga).

Preprint submitted to Elsevier

19 December 2007

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(Sλ )

−∆

qv

= λFu (x, u, v) in Ω = λFv (x, u, v) in Ω,

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−∆p u

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Costa [8], Dinu [9], [10], Fan and Zhao [11] Felmer, Man´asevich, and de Th´elin [12], de Figueiredo [13], Grossinho [15], and Krist´aly [17–19]. In order to be more precise, let us consider the following system of eigenvalue problems

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where Ω is a smooth (bounded or unbounded) domain in RN , F ∈ C 1 (Ω × R2 , R), p, q > 1, λ > 0 is a parameter, Fz denotes the partial derivative of F with respect to z, and ∆α is the α−Laplacian operator ∆α u = div(|∇u|α−2∇u). The number and certain properties of the solutions of (Sλ ) strongly depend not only on the shape of the domain Ω but also on the behavior of the function F at the origin and at infinity. Concerning the latter fact, the authors usually require the subcriticality and a certain superlinearity condition of F at infinity, as well as the superlinearity of F at the origin (see [3], [5], [13], and [18]).

(S λ,u0 ,v0 )

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The first aim of the present paper is to remove completely the above hypothesis. To handle this situation, we will use a recent idea due to B. Ricceri. The price we have to pay for this step is that we obtain “shifted” solutions for the system (Sλ ). Roughly speaking, under certain assumptions on the nonlinear term F and on Ω we are able to show the existence of a real parameter λ > 0 and of a pair (u0 , v0 ) ∈ W01,p (Ω) × W01,q (Ω) (where p, q > 1) such that the problem ′ −∆p (kukp (q−1) u) 1,p

−∆

q ′ (p−1)

q (kvk1,q

= λFu (x, u + u0 , v + v0 ) in Ω

u) = λFv (x, u + u0 , v + v0 ) in Ω,

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has at least three solutions. Here k · k1,α denotes the norm on W01,α (Ω) (for α ∈ {p, q}), while p′ and q ′ are the conjugates of p and q, respectively.

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The second aim of our paper is to treat also the case where F is not necessarily a C 1 function. For this we shall assume that F : R2 → R is only a locally Lipschitz function satisfying a certain regularity condition. In this case problem (S λ,u0 ,v0 ) (respectively, (Sλ )) will require a new formulation by means of hemivariational inequalities. Also, our study will be carried out in a more general setting than that of the Sobolev spaces W01,p (Ω) and W01,q (Ω). To define the systems we are going to investigate, let Ω be an open and unbounded subset of RN , p, q > 1, (X, || · ||X ) and (Y, || · ||Y ) be separable, uniformly convex, and smooth real Banach spaces which are compactly embedded in some spaces Lr (Ω), respectively, Ls (Ω) (with r ≥ p, s ≥ q), F : R2 → R be a locally Lipschitz regular function whose generalized directional derivatives satisfy certain conditions (involving p and q), and b: Ω → [0, +∞[ with b ∈ L1 (Ω) ∩ L∞ (Ω). For (u0 , v0 ) ∈ X × Y and λ > 0 we denote by S(u0 ,v0 ,λ) the problem of finding (u, v) ∈ X × Y such that for every (w1 , w2 ) ∈ X × Y the following inequalities 2

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hold

(A2 (v − v0 ))(w2 ) + λ ||v − v0 ||pq−q Y

R

Ω

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b(x)F2◦ (u(x), v(x); −w2 (x))dx ≥ 0,

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S(u0 ,v0 ,λ)

R pq−p ◦ ||u − u0 ||X (A1 (u − u0 ))(w1 ) + λ Ω b(x)F1 (u(x), v(x); −w1 (x))dx ≥ 0

where A1 [resp., A2 ] is the duality mapping on the space X [resp., Y ] induced by the weight function

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t ∈ [0, +∞[ 7→ tp−1 ∈ [0, +∞[ [resp., t ∈ [0, +∞[ 7→ tq−1 ∈ [0, +∞[],

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and F1◦ (u(x), v(x); −w1 (x)) [resp., F2◦ (u(x), v(x); −w2 (x))] denotes the (partial) generalized directional derivative of F (·, v(x)) [resp., F (u(x), ·)] at the point u(x) [resp., v(x)] in the direction −w1 (x) [resp., −w2 (x)]. The main result of the paper states that, under suitable hypotheses, there exist (u0 , v0 ) ∈ X × Y and λ > 0 such that the problem S(u0 ,v0 ,λ) has at least three solutions in X × Y . The methods we have used to get this result are based on the link between best approximation and the non-smooth critical point theory, as performed in the recent works of Tsar’kov [26] and Ricceri [25]. More exactly, we have adapted the method used in [25] to the system S(u0 ,v0 ,λ) . However, it turned out that the case p = q is easier to handle as the general case p 6= q: Thus, in a foregoing paper (see [4]), we were able to treat the case p = q = 2 under the assumption that X and Y are Hilbert spaces in a very satisfactory manner. The key point for the adaption of Ricceri’s method for the general case p 6= q lies in considering an appropriate norm on X ×Y (namely a homogenized Minkowski type norme) which leads to the following energy function E(u0 ,v0 ,λ) : X × Y → R attached to the above defined problem S(u0 ,v0 ,λ) E(u0 ,v0 ,λ) (u, v) =

pq ||u − u0 ||pq X + ||v − v0 ||Y − λJ(u, v), pq

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where J: X × Y → R is defined by Z

b(x)F (u(x), v(x))dx.

Ω

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J(u, v) =

The final part of the paper is devoted to an application of our main theorem to certain hemivariational inequalities defined on the Sobolev spaces W01,p (Ω) and W01,q (Ω), where Ω is a strip-like domain in RN . (In this case the duality mappings A1 and A2 correspond to the p and q-Laplacian, respectively.) Despite the fact that the space W01,α (Ω) (with α > 1) cannot be compactly embedded in any space Lr (Ω), we are able to prove the existence of multiple solutions for our system by involving the principle of symmetric criticality. 3

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Prerequisites

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Throughout this section let (X, k · k) be a real normed space and (X ⋆ , k · k∗ ) its topological dual. We denote by h·, ·i the duality pairing between X ⋆ and X. Following Chabrowski [6], we introduce in the next result the so-called duality mapping A: X → X ∗ on the space X induced by the weight function t ∈ [0, +∞[ 7→ tp−1 ∈ [0, +∞[, where p > 1 is a fixed real.

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Theorem 2.1 Let X be a smooth real normed space and p > 1 a real number. Then there exists a unique map A: X → X ∗ satisfying for every x ∈ X the following conditions

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(DM1) hA(x), xi = ||A(x)||∗||x||, (DM2) ||A(x)||∗ = ||x||p−1. Moreover, the map A has the following properties:

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(i) hA(x) − A(y), x − yi ≥ (||x||p−1 −||y||p−1)(||x||−||y||), for every x, y ∈ X. p (ii) The functional x ∈ X 7→ ||x|| ∈ R is Gˆateaux differentiable having the p derivative A. For the sake of completeness we next briefly recall a few basic facts concerning the theory of generalized differentiation of locally Lipschitz functions (for more details we refer, for example, to [7] or [22]). Assume now that X is a Banach space. Definition 1 A function f : X → R is said to be locally Lipschitz if, for every x ∈ X, there exist a neighborhood U of x and a constant L > 0 such that |f (y) − f (z)| ≤ Lky − zk f or all y, z ∈ U.

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Definition 2 Let f : X → R be a locally Lipschitz function. The generalized directional derivative of f at the point x ∈ X in the direction y ∈ X is defined as f (z + τ y) − f (z) . f ◦ (x; y) := lim sup τ z→x, τ →0+ The generalized gradient of f at x ∈ X is the set ∂f (x) := {x⋆ ∈ X ⋆ : hx⋆ , yi ≤ f ◦ (x; y) for all y ∈ X}.

Remark. Let f : X → R be a locally Lipschitz function and x ∈ X. It can be shown that f ◦ (x; y) ∈ R for every y ∈ X. Also, the functional f ◦ (x, ·): X → R is subadditive and positively homogeneous, and there exists a real number 4

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L > 0 such that |f ◦ (x; y)| ≤ L||y|| for every y ∈ X. Thus, due to the Hahn– Banach theorem, the set ∂f (x) is nonempty.

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The next proposition summarizes the main properties of the generalized directional derivative which will be used in the sequel:

Proposition 2.1 Let f, g : X → R be locally Lipschitz functions. Then the following assertions hold:

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f ◦ (x; y) = max{hξ, yi : ξ ∈ ∂f (x)}, for every x, y ∈ X. (f + g)◦(x; y) ≤ f ◦ (x; y) + g ◦ (x; y), for every x, y ∈ X. (−f )◦ (x; y) = f ◦ (x; −y), for every x, y ∈ X. (Lebourg’s mean value theorem) For every x, y ∈ X there exist an element u on the open line segment joining x and y, and z ∈ ∂f (u) such that f (y) − f (x) = hz, y − xi.

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(a) (b) (c) (d)

Proof. (a) See assertion (b) of Proposition 2.1.2 of [7]. (b) See the proof of Proposition 2.3.3 of [7].

(c) See assertion (c) of Proposition 2.1.1 of [7].

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(d) See Theorem 2.3.7 of [7]. 2

Definition 3 The locally Lipschitz function f : X → R is said to be regular at the point x ∈ X if, for every y ∈ X, the following conditions are satisfied: (i) The one-sided directional derivative f ′ (x; y) := lim+ τ →0

f (x + τ y) − f (x) τ

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exists and is a real number. (ii) The equality f ′ (x; y) = f ◦ (x; y) holds.

The function f is called regular if it is regular at every point x ∈ X.

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Proposition 2.2 If f : X → R is a locally Lipschitz convex function then f is regular. Proof. See assertion (b) of Proposition 2.3.6 of [7]. 2 Notation. Let X1 , X2 be Banach spaces and f : X1 × X2 → R a locally Lipschitz function. Consider (u, v), (w, z) ∈ X1 × X2 . Put fv := f (·, v) and fu := f (u, ·). We denote by f1◦ (u, v; w) [f2◦ (u, v; z)] 5

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the (partial) generalized directional derivative of fv [fu ] at the point u [v] in the direction w [z]. Also,

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∂1 f (u, v) and ∂2 f (u, v)

stay for the (partial) generalized gradient of fv at u, respectively, of fu at v.

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Proposition 2.3 Let X1 , X2 be Banach spaces and f : X1 × X2 → R a locally Lipschitz function which is regular at (u, v) ∈ X1 × X2 . Then the following assertions hold: (a) ∂f (u, v) ⊆ ∂1 f (u, v) × ∂2 f (u, v). (b) f ◦ ((u, v); (w, z)) ≤ f1◦ (u, v; w) + f2◦ (u, v; z), for every (w, z) ∈ X1 × X2 .

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Proof. (a) See Proposition 2.3.15 of [7]. Assertion (b) follows from assertion (a) of Proposition 2.1 and the foregoing statement. 2 Definition 4 Let f : X → R be a locally Lipschitz function. A point x ∈ X is a critical point of f , if 0 ∈ ∂f (x), that is, f ◦ (x; y) ≥ 0 for all y ∈ X.

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Remark. Note that every local extremum of f is a critical point of f . Definition 5 The function f : X → R satisfies the Palais–Smale condition if every sequence (xn ) in X satisfying the conditions (P S1 ) (f (xn )) is bounded, (P S2 ) there exists a sequence (εn ) in ]0, +∞[ with εn → 0 such that f ◦ (xn ; y − xn ) + εn ky − xn k ≥ 0 for all (y, n) ∈ X × N,

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admits a convergent subsequence.

We can state now the following mountain pass theorem:

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Theorem 2.2 Let f : X → R be a locally Lipschitz function satisfying the Palais–Smale condition, and x, y two different local minima of f . Then f has a (third) critical point in X different from x and y. Proof. The assertion follows from Corollary 3.2 of [23]. 2 We conclude this section by recalling two results which will be used in the proof of our main theorem. The first is a topological minimax theorem: Theorem 2.3 (Theorem 1 and Remark 1 of [24]) Let X be a topological space, Λ a real interval, and f : X × Λ → R a function satisfying the following 6

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conditions:

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(i) For every x ∈ X, the function f (x, ·) is quasi–concave and continuous. (ii) For every λ ∈ Λ, the function f (·, λ) is lower semicontinuous and each of its local minima is a global minimum. (iii) There exist ρ0 > supΛ inf X f and λ0 ∈ Λ such that {x ∈ X : f (x, λ0 ) ≤ ρ0 } is compact. Then Λ

X

X

Λ

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sup inf f = inf sup f.

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The second is a result from the theory of best approximation which, roughly speaking, assures that, in a certain class of Banach spaces, every sequentially weakly closed Chebyshev set is convex: Theorem 2.4 (Theorem 2 of [26], Lemma 1 of [14]) Let X be a uniformly convex Banach space with strictly convex topological dual, M a sequentially weakly closed non–convex subset of X. Then, for any convex dense subset S of X, there exists x0 ∈ S such that the set {y ∈ M : ky − x0 k = d(x0 , M)}

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contains at least two points.

The main results

Throughout this section we are working under the following hypotheses:

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(C1) N is a natural number, Ω an open and unbounded subset of RN , and p, q > 1. (C2) (X, || · ||X ) is a separable, uniformly convex, and smooth real Banach space with the property that there exists r ≥ p such that X ⊂ Lr (Ω) and the inclusion map iX : (X, || · ||X ) → (Lr (Ω), || · ||r ) is compact. (C3) (Y, ||·||Y ) is a separable, uniformly convex, and smooth real Banach space with the property that there exists s ≥ q such that Y ⊂ Ls (Ω) and the inclusion map iY : (Y, || · ||Y ) → (Ls (Ω), || · ||s ) is compact. (C4) F : R × R → R is a regular and locally Lipschitz function with F (0, 0) = 0 and such that there exist real numbers k > 0, p1 ∈ ]0, p − 1[, and q1 ∈ ]0, q − 1[ with the following properties: (i) For every u, v ∈ R and every ξ1 ∈ ∂1 F (u, v) the inequality |ξ1 | ≤ k|u|p1 holds. (ii) For every u, v ∈ R and every ξ2 ∈ ∂2 F (u, v) the inequality |ξ2 | ≤ k|v|q1 holds.

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(C5) b: Ω → [0, +∞[ belongs to L1 (Ω) ∩ L∞ (Ω) and is not identically zero. (Note that b ∈ Lν (Ω), for every ν ≥ 1.)

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Also, in the sequel we will use the following notations:

(N1) A1 : X → X ∗ stays for the duality mapping on X induced by the weight function t ∈ [0, +∞[ 7→ tp−1 ∈ [0, +∞[, and A2 : Y → Y ∗ for the duality mapping on Y induced by

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t ∈ [0, +∞[ 7→ tq−1 ∈ [0, +∞[.

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(N2) ||·||: X ×Y → R denotes the (homogenized Minkowski type) norm defined by 1 pq pq ||(u, v)|| = (||u||pq X + ||v||Y ) , for every (u, v) ∈ X × Y. Throughout this section the space X × Y is considered endowed with this norm. By the properties of X and Y mentioned in (C2) and (C3), the space X × Y is a separable, uniformly convex, and smooth real Banach space. Note that the following inequality holds for every (u, v) ∈ X × Y

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||u||X + ||v||Y ≤ 2||(u, v)||.

(1)

(N3) According to conditions (C2) and (C3), there exists a positive real number c such that the following inequalities hold for every (u, v) ∈ X × Y ||u||r ≤ c||u||X , (N4) ν1 :=

r , r−(p1 +1)

||v||s ≤ c||v||Y .

ν2 :=

(2)

s . s−(q1 +1)

Remark. If p = q one can consider a simpler norm on X × Y , for example 1

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||(u, v)|| = (||u||pX + ||v||pY ) p .

Lemma 3.1 The following inequality holds for every (x, y), (¯ x, y¯) ∈ R × R

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|F (x, y) − F (¯ x, y¯)| ≤ k(|x| + |¯ x|)p1 |x − x¯| + k(|y| + |¯ y |)q1 |y − y¯|. Proof. Let (x, y), (¯ x, y¯) ∈ R × R. According to assertion (d) of Proposition 2.1 and to assertion (a) of Proposition 2.3, there exist a pair (u, v) on the open line segment joining (x, y) and (¯ x, y¯), ξ1 ∈ ∂1 F (u, v), and ξ2 ∈ ∂2 F (u, v) such that F (x, y) − F (¯ x, y¯) = ξ1 (x − x¯) + ξ2 (y − y¯). Using condition (C4), we conclude that |F (x, y) − F (¯ x, y¯)| ≤ |ξ1 (x − x¯)| + |ξ2(y − y¯)| ≤ k|u|p1 |x − x¯| + k|v|q1 |y − y¯|. 8

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Since (u, v) belongs to the line segment determined by (x, y) and (¯ x, y¯), we get the asserted inequality |F (x, y) − F (¯ x, y¯)| ≤ k(|x| + |¯ x|)p1 |x − x¯| + k(|y| + |¯ y |)q1 |y − y¯|. 2

x ∈ Ω 7−→ b(x)F (u(x), v(x)) ∈ R belongs to L1 (Ω).

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Lemma 3.2 For every pair (u, v) ∈ Lr (Ω) × Ls (Ω) the map

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Proof. By Lemma 3.1, the following inequality holds for every x ∈ Ω (recall that F (0, 0) = 0) |F (u(x), v(x))| ≤ k|u(x)|p1+1 + k|v(x)|q1 +1 .

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Thus

|b(x)F (u(x), v(x))| ≤ k|b(x)||u(x)|p1 +1 + k|b(x)||v(x)|q1 +1 . Since

1 + ν1

1

r p1 +1

= 1,

1 + ν2

s

1

s q1 +1

(3) r

= 1, b ∈ Lν1 (Ω)∩Lν2 (Ω), |u|p1+1 ∈ L p1 +1 (Ω), and

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since |v|q1+1 ∈ L q1 +1 (Ω), we conclude that |b||u|p1+1 ∈ L1 (Ω) and |b||v|q1+1 ∈ L1 (Ω). According to (3), this finishes the proof. 2 In view of Lemma 3.2 we can now define the map J: X × Y → R by J(u, v) =

Z

b(x)F (u(x), v(x))dx.

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Proposition 3.1 The map J defined above has the following properties: (a) There exists a positive real number C such that the inequality

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p1+1 |J(u, v)| ≤ C(||u||X + ||v||qY1+1 )

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holds for every pair (u, v) ∈ X × Y . (b) J is locally Lipschitz. (c) J is sequentially weakly continuous. (d) For every (u, v), (w1, w2 ) ∈ X × Y the map x ∈ Ω 7−→ b(x)F ◦ ((u(x), v(x)); (w1(x), w2 (x))) ∈ R

belongs to L1 (Ω) and the following inequalities hold J ◦ ((u, v); (w1, w2 )) ≤

Z

b(x)F ◦ ((u(x), v(x)); (w1 (x), w2 (x)))dx

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(4)

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and (5)

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J ◦ ((u, v); (w1, w2 )) ≤ k(||b||ν1 ||u||pr 1 ||w1||r + ||b||ν2 ||v||qs1 ||w2 ||s ).

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Proof. (a) Using (3) and H¨older’s inequality, we get that

|J(u, v)| ≤ k||b||ν1 ||u||rp1+1 + k||b||ν2 ||v||qs1+1 , hence, in view of (2),

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p1+1 |J(u, v)| ≤ k||b||ν1 cp1 +1 ||u||X + k||b||ν2 cq1 +1 ||v||qY1+1 .

Taking C = max{k||b||ν1 cp1 +1 , k||b||ν2 cq1 +1 }, we obtain the asserted inequality.

|J(u, v) − J(¯ u, v¯)| ≤ k

Z

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(b) We prove that J is Lipschitz on bounded sets. Let M ≥ 0 and (u, v), (¯ u, v¯) ∈ X × Y be so that ||u||X , ||¯ u||X , ||v||Y , ||¯ v||Y ≤ M. In view of Lemma 3.1 the following inequality holds b(x)(|u(x)| + |¯ u(x)|)p1 |u(x) − u¯(x)|dx +

Ω

k

Z

b(x)(|v(x)| + |¯ v(x)|)q1 |v(x) − v¯(x)|dx.

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Ω

Using H¨older’s inequality (note that further get that

1 ν1

+

p1 r

+

1 r

= 1 and

1 ν2

+

q1 s

+

1 s

= 1), we

u|)||pr 1 ||u − u |J(u, v) − J(¯ u, v¯)| ≤ k||b||ν1 ||(|u| + |¯ v|)||qs1 ||v − v¯||s . ¯||r + k||b||ν2 ||(|v| + |¯

(6)

Applying the subadditivity of the norms || · ||r and || · ||s, the above inequality yields, according to (2) and to the assumption that ||u||X , ||¯ u||X , ||v||Y , ||¯ v||Y are ≤ M,

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|J(u, v)−J(¯ u, v¯)| ≤ k2p1 cp1 +1 M p1 ||b||ν1 ||u− u¯||X +k2q1 cq1 +1 M q1 ||b||ν2 ||v − v¯||Y . Taking L = max{k2p1 ||b||ν1 cp1 +1 M p1 , k2q1 ||b||ν2 cq1 +1 M q1 } and using (1), we finally obtain, that

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|J(u, v) − J(¯ u, v¯)| ≤ L(||u − u¯||X + ||v − v¯||Y ) ≤ 2L||(u, v) − (¯ u, v¯)||.

(c) Let (un , vn )n∈N be a sequence in X × Y which converges weakly to (u, v) ∈ X ×Y . It follows that (un )n∈N converges weakly to u in (X, ||·||X ), and (vn )n∈N converges weakly to v in (Y, || · ||Y ). Then (un )n∈N and (vn )n∈N are bounded, hence we find a real number M ≥ 0 such that ||un ||X , ||u||X , ||vn ||Y , ||v||Y ≤ M, for every n ∈ N. Since a compact linear operator A: V → W between normed spaces is sequentially weakly continuous (if W carries the norm-topology), we 10

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conclude that (un )n∈N converges to u in the space (Lr (Ω), || · ||r ) and (vn )n∈N converges to v in the space (Ls (Ω), ||·||s). From (6) we obtain that the following inequality holds for every index n ∈ N

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|J(un , vn ) − J(u, v)| ≤ k2p1 cp1 M p1 ||b||ν1 ||un − u||r + k2q1 cq1 M q1 ||b||ν2 ||vn − v||s , hence lim J(un , vn ) = J(u, v).

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(d) Pick (u, v), (w1, w2 ) ∈ X × Y . According to assertion (a) of Proposition 2.1 and to assertion (a) of Proposition 2.3, for every x ∈ Ω, there exist ξ1 ∈ ∂1 F (u(x), v(x)) and ξ2 ∈ ∂2 F (u(x), v(x)) such that F ◦ ((u(x), v(x)); (w1(x), w2 (x))) = ξ1 (w1 (x)) + ξ2 (w2 (x)).

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Using condition (C4), we get that

|F ◦ ((u(x), v(x)); (w1(x), w2 (x)))| ≤ k|u(x)|p1 |w1(x)| + k|v(x)|q1 |w2 (x)|.(7) 1 + pr1 + 1r ν1

= 1 and

1 + qs1 + 1s ν2

= 1)

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Since b|u|p1 |w1 |, b|v|q1 |w2 | ∈ L1 (Ω) (note that we conclude that the map

x ∈ Ω 7−→ b(x)F ◦ ((u(x), v(x)); (w1(x), w2 (x))) ∈ R belongs to L1 (Ω). For inequality (4) we refer to section 5.1, pages 134–138, of [20]. To prove now (5), we start from (7). By an application of H¨older’s inequality we get that

ZΩ

b(x)F ◦ ((u(x), v(x)); (w1(x), w2 (x)))dx ≤ |b(x)F ◦ ((u(x), v(x)); (w1(x), w2 (x)))|dx ≤

Ω

EP

Z

k(||b||ν1 ||u||pr 1 ||w1 ||r + ||b||ν2 ||v||qs1 ||w2 ||s ).

AC C

Together with (4) the above inequality yields (5). 2 For (u0 , v0 ) ∈ X × Y and λ > 0 we denote by S(u0 ,v0 ,λ) the problem of finding (u, v) ∈ X × Y such that for every (w1 , w2 ) ∈ X × Y the following inequalities hold

S(u0 ,v0 ,λ)

R pq−p ◦ ||u − u0 ||X (A1 (u − u0 ))(w1 ) + λ Ω b(x)F1 (u(x), v(x); −w1 (x))dx ≥ 0

(A2 (v − v0 ))(w2 ) + λ ||v − v0 ||pq−q Y

11

R

Ω

b(x)F2◦ (u(x), v(x); −w2 (x))dx ≥ 0.

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pq ||u − u0 ||pq X + ||v − v0 ||Y − λJ(u, v). pq

RI P

E(u0 ,v0 ,λ) (u, v) =

T

Also, let E(u0 ,v0 ,λ) : X × Y → R be the energy functional attached to the above defined problem S(u0 ,v0 ,λ) , i.e.,

Note that this map is locally Lipschitz.

SC

Lemma 3.3 If (u, v) is a critical point of E(u0 ,v0 ,λ) , then (u, v) is a solution of the problem S(u0 ,v0 ,λ) .

AN U

Proof. Consider an arbitrary pair (w1 , w2 ) ∈ X × Y . Since (u, v) is a critical point of E(u0 ,v0 ,λ) , assertion (b) of Proposition 2.1, Proposition 2.2, and assertion (ii) of Theorem 2.1 yields that ◦ 0 ≤ E(u ((u, v); (w1, w2)) ≤ ||u − u0 ||pq−p (A1 (u − u0 ))(w1 ) + X 0 ,v0 ,λ)

||v − v0 ||pq−q (A2 (v − v0 ))(w2 ) + λ(−J)◦ ((u, v); (w1, w2 )). Y Taking into account assertion (c) of Proposition 2.1, we further obtain that

By (4) we get that

TE DM

(A2 (v−v0 ))(w2 )+λJ ◦ ((u, v); (−w1, −w2 )). (A1 (u−u0 ))(w1 )+||v−v0 ||pq−q 0 ≤ ||u−u0||pq−p Y X

0 ≤ ||u − u0 ||pq−p (A1 (u − u0 ))(w1 ) + ||v − v0 ||pq−q (A2 (v − v0 ))(w2 ) + X Y λ

Z

b(x)F ◦ ((u(x), v(x)); (−w1(x), −w2 (x)))dx

Ω

Assertion (b) of Proposition 2.3 finally yields that

λ

Z

Ω

EP

0 ≤ ||u − u0 ||pq−p (A1 (u − u0 ))(w1 ) + ||v − v0 ||pq−q (A2 (v − v0 ))(w2 ) + X Y b(x)(F1◦ (u(x), v(x); −w1(x)) + F2◦ (u(x), v(x); −w2 (x)))dx.

AC C

Choosing in the above inequality w1 = 0, respectively, w2 = 0, the assertion follows. 2 Proposition 3.2 Let (u0 , v0 ) ∈ X × Y and λ > 0. The energy functional E(u0 ,v0 ,λ) satisfies the following conditions: (a) E(u0 ,v0 ,λ) is coercive. (b) E(u0 ,v0 ,λ) verifies the Palais-Smale condition. (c) E(u0 ,v0 ,λ) is weakly lower semicontinuous.

12

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T

Proof. (a) According to assertion (a) of Proposition 3.1, the following inequality holds for every (u, v) ∈ X × Y

RI P

||v − v0 ||pq ||u − u0 ||pq Y X − λC||u||pX1+1 + − λC||v||Yq1+1 ≤ E(u0 ,v0 ,λ) (u, v). (8) pq pq Define g: X → R and h: Y → R by

SC

||u − u0 ||pq ||v − v0 ||pq p1+1 X Y g(u) = − λC||u||X and h(v) = − λC||v||Yq1+1 . pq pq With these notations (8) can be rewritten as

For u 6= 0 we have that

= ||u||pX Since lim||u||X →∞

||u − u0 ||pq p1+1−p X − λC||u||X p pq||u||X ! ||u − u0 ||pX pq−p p1+1−p ||u − u0 ||X − λC||u||X . pq||u||pX !

TE DM

g(u) = ||u||pX

(9)

AN U

g(u) + h(v) ≤ E(u0 ,v0 ,λ) (u, v), for all (u, v) ∈ X × Y.

||u−u0 ||pX ||u||pX

= +∞ (because pq − = 1, lim||u||X →∞ ||u − u0 ||pq−p X

p1 +1−p = 0 (recall that p1 < p − 1), we obtain that p > 0), and lim||u||X →∞ ||u||X lim||u||X →∞ g(u) = +∞. It follows that g is coercive. A similar argument yields that h is coercive, too. Since g and h are continuous it follows from (9) that E(u0 ,v0 ,λ) is coercive.

AC C

EP

(b) Let (un , vn ) be a sequence in X × Y satisfying the conditions (P S1) and (P S2) of Definition 5 for the function E(u0 ,v0 ,λ) . According to the coercivity of the map E(u0 ,v0 ,λ) , the sequence (un , vn ) is bounded. Thus (by the reflexivity of X × Y ) this sequence has a weakly convergent subsequence. Without any loss of generality we can assume that the sequence (un , vn ) converges weakly to (u, v) ∈ X × Y , i.e., (un ) converges weakly to u and (vn ) converges weakly to v. In view of condition (P S2) the following inequality holds for every natural number n ||un − u0 ||pq−p (A1 (un − u0 ))(u − un ) + ||vn − v0 ||pq−q (A2 (vn − v0 ))(v − vn ) X Y ◦ +λJ ((un , vn ); (un − u, vn − v)) + εn ||(u − un , v − vn )|| ≥ 0.

Put

tn := λk(||b||ν1 ||un ||pr 1 ||un − u||r + ||b||ν2 ||vn ||qs1 ||vn − v||s ) + εn ||(u − un , v − vn )||, 13

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αn := ||un − u0 ||pq−p (A1 (un − u0 ))(un − u), X

T

βn := ||vn − v0 ||pq−q (A2 (vn − v0 ))(vn − v). Y

RI P

Using these notations and (5), the above inequality yields that for every natural number n tn ≥ αn + βn .

(10)

Put x0 := ||u − u0 ||X and y0 := ||v − v0 ||Y . Define f, g: [0, +∞[ by

g(x) = xpq−q (xq−1 − y0q−1)(x − y0 ).

SC

f (x) = xpq−p (xp−1 − xp−1 0 )(x − x0 ),

AN U

Note that f (x), g(x) ≥ 0 for every x ≥ 0. According to assertion (i) of Theorem 2.1, the following inequalities hold for every natural number n αn − ||un − u0||pq−p (A1 (u − u0))(un − u) ≥ f (||un − u0 ||X ), X pq−q βn − ||vn − v0 ||Y (A2 (v − v0 ))(vn − v) ≥ g(||vn − v0 ||Y ), thus, by (10),

TE DM

tn − ||un − u0 ||pq−p (A1 (u − u0 ))(un − u) − X pq−q ||vn − v0 ||Y (A2 (v − v0 ))(vn − v) ≥ f (||un − u0 ||X ) + g(||vn − v0 ||Y ).

EP

The sequences (||un ||X ) and (||vn ||Y ) are bounded, and, by conditions (C2) and (C3), we know that lim ||un − u||r = lim ||vn − v||s = 0, also (since (un , vn )n converges weakly to (u, v)) we have lim(A1 (u − u0 ))(un − u) = lim(A2 (v − v0 ))(vn − v) = 0, hence the limit (when n → ∞) of the left side of the above inequality is 0. It follows that lim f (||un − u0 ||X ) = lim g(||vn − v0 ||Y ) = 0, thus lim ||un − u0 ||X = x0 = ||u − u0 ||X and lim ||vn − v0 ||Y = y0 = ||v − v0 ||Y . Since X and Y are uniformly convex, we conclude that lim ||un − u||X = lim ||vn − v||Y = 0, thus E(u0 ,v0 ,λ) has the Palais-Smale condition. pq

AC C

0 ,v0 )|| (c) The map (u, v) ∈ X×Y 7−→ ||(u,v)−(u ∈ R is weakly lower semicontinpq uous since all its lower level sets are weakly closed (recall that X ×Y is reflexive since it is uniformly convex). The map (u, v) ∈ X × Y 7−→ −λJ(u, v) ∈ R is sequentially weakly continuous (according to assertion (c) of Proposition 3.1), hence the map E(u0 ,v0 ,λ) is sequentially weakly lower semicontinuous. Since E(u0 ,v0 ,λ) is coercive, the Eberlein-Smulyan theorem finally implies that E(u0 ,v0 ,λ) is weakly lower semicontinuous. 2

Theorem 3.1 (The main result) Assume that the hypotheses (C1)–(C5) hold, and that the function J is not constant. Then, for every σ ∈] inf J, sup J[ X×Y

X×Y

and every (u0 , v0 ) ∈ J −1 (] − ∞, σ[), one of the following alternatives is true: 14

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RI P

T

(A1 ) There exists λ > 0 such that the function E(u0 ,v0 ,λ) has at least three critical points in X × Y . (A2 ) There exists (u∗ , v ∗ ) ∈ J −1 (σ) such that, for all (u, v) ∈ J −1 ([σ, +∞[) \ {(u∗ , v ∗ )}, the inequality k(u, v) − (u0 , v0 )k > ||(u∗, v ∗ ) − (u0 , v0 )||. holds.

AN U

SC

Proof. Fix σ ∈] inf X×Y J, supX×Y J[, (u0 , v0 ) ∈ J −1 (] − ∞, σ[), and assume that (A1 ) does not hold. Note that this implies that for every λ > 0 the function E(u0 ,v0 ,λ) has at most one local minimum in X × Y . Indeed, the existence of two local minima of E(u0 ,v0 ,λ) would imply, by Theorem 2.2 (recall that, according to assertion (b) of Proposition 3.2, the locally Lipschitz function E(u0 ,v0 ,λ) satisfies the Palais-Smale condition) that this function has a third critical point. This would contradict our assumption that (A1 ) does not hold. We are going to prove that in this case (A2 ) is satisfied. For this put Λ := [0, +∞[ and define the map f : X × Y × Λ → R by ||(u, v) − (u0 , v0 )||pq + λ(σ − J(u, v)). pq

TE DM

f (u, v, λ) = E(u0 ,v0 ,λ) (u, v) + λσ =

EP

We consider the space X × Y be equipped with the weak topology. Our aim is to show that f satisfies the conditions (i), (ii), and (iii) of Theorem 2.3. Observe first that for every (u, v) ∈ X × Y the map f (u, v, ·): Λ → R is afine, thus it is quasi-concave and continuous, i.e., condition (i) is fulfilled. Next we prove that condition (ii) is also satisfied. For this fix a real number λ ≥ 0. By assertion (c) of Proposition 3.2 we know that f (·, λ) is (weakly) lower semicontinuous. Assume now that f (·, λ) has a local minimum which is not global. Since f (·, λ) is coercive (by assertion (a) of Proposition 3.2) and weakly lower semicontinuous it has a global minimum. Thus, f (·, λ) has at least two local minima. It follows that E(u0 ,v0 ,λ) has at least two local minima, too, which is impossible by the assumption and the remark made at the beginning of the proof. We conclude that condition (ii) holds. To prove that condition (iii) is also satisfied we show first that sup inf f (u, v, λ) < +∞. For this choose Λ X×Y

AC C

(u1 , v1 ) ∈ X × Y such that σ < J(u1 , v1 ). For every λ ∈ Λ we have that inf f (u, v, λ) ≤ f (u1 , v1 , λ) ≤

X×Y

k(u1, v1 ) − (u0 , v0 )kpq , pq

(11)

hence sup inf f (u, v, λ) < +∞. For every ρ0 > sup inf f (u, v, λ) the set Λ X×Y

Λ X×Y

{(u, v) ∈ X × Y | f (u, v, 0) ≤ ρ0 } is weakly compact, thus condition (iii) is 15

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satisfied. Applying Theorem 2.3, we obtain that X×Y

(12)

Λ

RI P

Λ X×Y

T

α := sup inf f = inf sup f.

Note that the function λ ∈ Λ 7−→ inf f (u, v, λ) is upper semicontinuous X×Y

since for every (u, v) ∈ X × Y the map f (u, v, ·) is continuous. Also, the first inequality of (11) yields that lim inf f (u, v, λ) = −∞.

SC

λ→+∞ X×Y

Thus, the map λ ∈ Λ 7−→ inf f (u, v, λ) is upper bounded and attains its X×Y

maximum at a point λ∗ ∈ Λ. Hence

Λ X×Y

X×Y

AN U

α = sup inf f = inf

||(u, v) − (u0 , v0 )||pq + λ∗ (σ − J(u, v)) . pq !

(13)

On the other hand, we have for every (u, v) ∈ X × Y that

Λ

if σ > J(u, v)

TE DM

sup f (u, v, λ) =

+∞,

In view of (12) it follows that α=

inf

||(u,v)−(u0 ,v0 )||pq , pq

J −1 ([σ,+∞[)

if σ ≤ J(u, v).

||(u, v) − (u0 , v0 )||pq . pq !

pq

α=

EP

0 ,v0 )|| ∈ R is weakly lower semiSince the map (u, v) ∈ X × Y 7−→ ||(u,v)−(u pq continuous and coercive, and since the set J −1 ([σ, +∞[) is weakly closed and nonempty, there exists a pair (u∗ , v ∗) ∈ J −1 ([σ, +∞[) such that

||(u∗, v ∗ ) − (u0 , v0 )||pq . pq

(14)

AC C

Observe that λ∗ > 0. (Otherwise, λ∗ = 0. This implies α = 0, hence (u0 , v0 ) = (u∗ , v ∗ ), which is impossible since J(u0 , v0 ) < σ.) The relations (13) and (14) yield that ||(u∗, v ∗ ) − (u0 , v0 )||pq ||(u∗, v ∗ ) − (u0 , v0 )||pq ≤ + λ∗ (σ − J(u∗ , v ∗ )), pq pq

thus J(u∗ , v ∗ ) = σ. This implies (by (13)) that (u∗ , v ∗ ) is a global minimum of E(u0 ,v0 ,λ∗ ) . By the assumption and the remark at the beginning of 16

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T

the proof this pair is the only global minimum of E(u0 ,v0 ,λ∗ ) . Thus, for every pair (u, v) ∈ J −1 ([σ, +∞[) \ {(u∗ , v ∗ )}, the inequality k(u, v) − (u0 , v0 )k > ||(u∗, v ∗ ) − (u0 , v0 )|| holds. We conclude that (A2 ) is satisfied. 2

SC

RI P

Corollary 3.1 Assume that the hypotheses of Theorem 3.1 are fulfilled. If S is a convex dense subset of X × Y , and if there exists σ ∈] inf X×Y J, supX×Y J[ such that the level set J −1 ([σ, +∞[) is not convex, then there exist (u0, v0 ) ∈ J −1 (] − ∞, σ[) ∩ S and λ > 0 such that the function E(u0 ,v0 ,λ) has at least three critical points in X × Y , i.e., there exist (u0 , v0 ) ∈ J −1 (] − ∞, σ[) ∩ S and λ > 0 such that problem S(u0 ,v0 ,λ) has at least three solutions in X × Y .

AN U

Proof. Since J is sequentially weakly continuous (by assertion (c) of Proposition 3.1), the level set M := J −1 ([σ, +∞[) is sequentially weakly closed. Theorem 2.4 yields now the existence of pairwise distinct pairs (u0 , v0 ) ∈ S, (u1 , v1 ), (u2 , v2 ) ∈ M such that ||(u1 , v1 ) − (u0 , v0 )|| = ||(u2 , v2 ) − (u0, v0 )|| =

inf

(u,v)∈M

||(u, v) − (u0 , v0 )||.

TE DM

It follows that (u0 , v0 ) ∈ / M, i.e., J(u0 , v0 ) < σ. Also, the above relations show that alternative (A2 ) of Theorem 3.1 does not hold. Hence (A1 ) must be satisfied, i.e., there exists λ > 0 such that the function E(u0 ,v0 ,λ) has at least three critical points in X ×Y . Using Lemma 3.3, we conclude that the problem S(u0 ,v0 ,λ) has at least three solutions in X × Y . 2 Remark. The assumption of Corollary 3.1 stating that there exists σ ∈ ] inf X×Y J, supX×Y J[ such that the level set J −1 ([σ, +∞[) is not convex is equivalent to the fact that J is not quasi-concave.

An application involving the principle of symmetric criticality

EP

4

We start by briefly recalling this principle:

AC C

Theorem 4.1 ([16]) Let X be a (real) Banach space, G a compact topological group, π: G × X → X a continuous action of G on X, X G := {x ∈ X | π(g, x) = x, ∀g ∈ G} the set of fixed points of the action π, and φ: X → R a locally Lipschitz and G-invariant function (i.e., φ(π(g, x)) = φ(x), for every (g, x) ∈ G × X). Then every critical point of φ|X G : X G → R is also a critical point of φ. Let N, m be natural numbers with m ≥ 1 and N − m ≥ 2. In this section we consider strip-like domains in RN of the form Ω := ω × RN −m , where ω ⊆ Rm is open and bounded, and 0 ∈ Ω. Throughout this section, for every α > 1, 17

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1/α

Z

kuk1,α = |∇u|α dx

T

we consider the Sobolev space W01,α (Ω) be endowed with the norm .

RI P

Ω

(A(u))(v) =

Z

SC

It is known that, for 1 < α < N, W01,α (Ω) becomes this way a separable, uniformly convex, and smooth real Banach space, and that (due to the cone α ), the Sobolev embedding property of Ω), for every β ∈ [α, α∗] (where α∗ = NN−α 1,α β W0 (Ω) ֒→ L (Ω) holds. Also, a straightforward computation yields that the map A: W01,α (Ω) → (W01,α (Ω))∗ defined by |∇u|α−2∇u∇vdx, for every u, v ∈ W01,α (Ω),

Ω

Consider now p, q ∈]1, N[ and put X := W01,p (Ω),

AN U

is the duality mapping induced by the function t ∈ [0, +∞[→ tα−1 ∈ [0, +∞[.

Y := W01,q (Ω).

TE DM

The product space X × Y will be endowed with the norm

pq k(u, v)k = kukpq 1,p + kvk1,q

1 pq

.

Let F : R2 → R be a map which satisfies condition (C4) and b: Ω → [0, +∞[ a function satisfying condition (C5). According to the above considerations, for (u0 , v0 ) ∈ X × Y and λ > 0, the problem S(u0 ,v0 ,λ) defined in the previous section has now the following form: We look for (u, v) ∈ X × Y such that for every (w1 , w2) ∈ X × Y the following inequalities hold |∇(u − u0 )|

p−2

∇(u − u0 )∇w1 dx + λ

EP

′ S(u 0 ,v0 ,λ)

Z pq−p ||u − u0 ||1,p Ω Z pq−q ||v − v || 0 1,q

|∇(v − v0 )|q−2 ∇(v − v0 )∇w2 dx + λ

Ω

Z

b(x)F1◦ (u(x), v(x); −w1 (x))dx ≥ 0

Ω

Z

b(x)F2◦ (u(x), v(x); −w2 (x))dx ≥ 0.

Ω

AC C

The map E(u0 ,v0 ,λ) : X × Y → R defined by pq ||u − u0 ||pq 1,p + ||v − v0 ||1,q − λJ(u, v) E(u0 ,v0 ,λ) (u, v) = pq

′ is the energy functional attached to the problem S(u . Recall that J: X × 0 ,v0 ,λ) Y → R is the map defined by

J(u, v) =

Z

b(x)F (u(x), v(x))dx.

Ω

18

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RI P

T

Taking into account the fact that the spaces W01,α (Ω) cannot be compactly embedded into any space Lβ (Ω) for β ∈ [α, α∗], we cannot apply directly the ′ results obtained in the previous section to the system S(u . To eliminate 0 ,v0 ,λ) this inconvenient, we consider for α ∈ {p, q} the continuous action of the compact Lie group G := O(N − m) of orthogonal linear transformations of RN −m on W01,α (Ω) defined by πα (g, u)(x, y) = u(x, g −1y), ∀ (g, u) ∈ G × W01,α (Ω), ∀ (x, y) ∈ ω × RN −m .

SC

The fixed points of this action are the so-called axially symmetric elements of W01,α (Ω). Clearly, G acts on X × Y by (g, (u, v)) ∈ G × X × Y 7−→ (πp (g, u), πq (g, v)) ∈ X × Y.

AN U

Also, (X × Y )G = X G × Y G .

TE DM

Corollary 4.1 Let X and Y be as above, F : R2 → [0, +∞[ a function which is not quasi-concave and which satisfies condition (C4), b: RN → [0, +∞[ a Ginvariant function (i.e., b(x, y) = b(x, gy) for every (g, x, y) ∈ G × ω × RN −m ) with b(0) > 0 and for which condition (C5) holds, and S a convex dense subset of X G × Y G . Then there exist (u0 , v0 ) ∈ S and λ > 0 with the property that ′ the problem S(u has at least three solutions in X G × Y G . 0 ,v0 ,λ)

EP

Proof. We are going to apply Corollary 3.1 to the spaces X G and Y G and to the map J|X G ×Y G . According to a result contained in [21], the spaces X G and Y G satisfy conditions (C2) and (C3) for r ∈]p, p∗ [ and s ∈]q, q ∗ [. First we prove that J|X G ×Y G is not constant. For this we assume, by contradiction, that J|X G ×Y G (u, v) = J|X G ×Y G (0, 0) = 0 for every (u, v) ∈ X G × Y G . Since b ≥ 0 and F ≥ 0, it follows that b(x)F (u(x), v(x)) = 0 for every (u, v) ∈ X G × Y G and a.e. x ∈ Ω. Since b(0) > 0, there exists a real number R > 0 such that the closed ball B centered in 0 with radius R is contained in Ω and b(x) > 0 for a.e. x ∈ B. Consequently, F (u(x), v(x)) = 0 for every (u, v) ∈ X G × Y G and a.e. x ∈ B. Since F is not constant, there exist s0 , t0 ∈ R such that F (s0 , t0 ) > 0. Choosing u ∈ X G and v ∈ Y G such that u(x) = s0 and v(x) = t0 for every x ∈ B, we get a contradiction.

AC C

Next we show that J|X G ×Y G is not quasi-concave. It follows from above that t :=

Z

b(x)dx > 0.

B

The fact that F is not quasi-concave implies the existence of a real number ρ0 ∈]0, sup F [ such that F −1 ([ρ0 , +∞[) is not convex. Thus we find ρ ∈ R, R×R

α ∈]0, 1[, and (si , ti ) ∈ R2 , i ∈ {1, 2, 3}, with the following properties (s2 , t2 ) = α(s1 , t1 ) + (1 − α)(s3 , t3 ), F (s1 , t1 ) > ρ, F (s3 , t3 ) > ρ, F (s2 , t2 ) < ρ. 19

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Let

T

M := max{F (x, y) | |x| ≤ max{|s1 |, |s3|}, |y| ≤ max{|t1 |, |t3 |}}.

RI P

Choose R1 > R and ε > 0 such that

kbk∞ Mmeas(A) < ε < t|F (si , ti ) − ρ|, for i ∈ {1, 2, 3},

0,

|x| ≤ R and vi (x) =

ti , 0,

|x| ≥ R1 ,

|x| ≤ R

|x| ≥ R1 .

AN U

ui (x) =

si ,

SC

where A := {x ∈ Ω : R < |x| < R1 } and meas(A) stays for the Lebesgue measure of A. For i ∈ {1, 3} let ui , vi ∈ Cc∞ (Ω) be G-invariant functions such that ||ui||∞ = |si |, ||vi ||∞ = |ti |,

Put (u2 , v2 ) := α(u1 , v1 ) + (1 − α)(u3 , v3 ). It follows that u2 , v2 ∈ Cc∞ (Ω), ||u2||∞ ≤ α|s1 | + (1 − α)|s3 |, ||v2 ||∞ ≤ α|t1 | + (1 − α)|t3 |,

u2 (x) =

|x| ≤ R

t2 ,

and v2 (x) =

TE DM

s2 , 0,

|x| ≥ R1 ,

0,

|x| ≤ R

|x| ≥ R1 .

Let i ∈ {1, 2, 3}. Note that for every x ∈ Ω the following inequalities hold |ui(x)| ≤ ||ui||∞ ≤ max{|s1 |, |s3 |} and |vi (x)| ≤ ||vi ||∞ ≤ max{|t1 |, |t3|}, thus

0 ≤ b(x)F (ui (x), vi (x)) ≤ ||b||∞ M.

EP

Since J|X G ×Y G (ui , vi ) =

Z

b(x)F (ui (x), vi (x))dx +

B

AC C

= F (si, ti )t +

Z

b(x)F (ui (x), vi (x))dx

A

Z

b(x)F (ui (x), vi (x))dx,

A

we conclude that, for i ∈ {1, 3}, J|X G ×Y G (ui , vi ) ≥ F (si, ti )t − ||b||∞Mmeas(A) > F (si , ti )t − ǫ > tρ.

and

J|X G ×Y G (u2 , v2 ) ≤ F (s2 , t2 )t + ||b||∞ Mmeas(A) < F (s2 , t2 )t + ǫ < tρ. 20

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Thus J|X G ×Y G is not quasi-concave. Therefore, Corollary 3.1 yields the existence of a pair (u0 , v0 ) ∈ S and of a real number λ > 0 such that the function E(u0 ,v0 ,λ) |X G ×Y G has at least three critical points in X G × Y G . In order to ensure that these points are also critical points of E(u0 ,v0 ,λ) we have to show that E(u0 ,v0 ,λ) is G-invariant. For this pick arbitrary (u, v) ∈ X × Y and g ∈ G. Then ||(g · u, g · v) − (u0 , v0 )||pq − λJ(g · u, g · v) pq Z ||(g · u, g · v) − (g · u0 , g · v0 )||pq = − λ b(x)F ((g · u)(x), (g · v)(x))dx. pq Ω

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E(u0 ,v0 ,λ) (g · u, g · v) =

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Using the formula for the change of variable, and taking into account that b is G-invariant and that the elements of G are orthogonal maps (hence the absolut value of the determinant of their matrices is 1), we conclude that ||(u, v) − (u0 , v0 )||pq −λ E(u0 ,v0 ,λ) (g · u, g · v) = pq

Z

b(x)F (u(x), v(x))dx

Ω

= E(u0 ,v0 ,λ) (u, v).

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So, by Theorem 4.1, every critical point of E(u0 ,v0 ,λ) |X G ×Y G is also a critical point of E(u0 ,v0 ,λ) . The conclusion follows now from Lemma 3.3. 2 Example. We give an example of a non-constant function F : R2 → [0, +∞[ which satisfies the conditions required in the hypotheses of Corollary 4.1, i.e., condition (C4) and not being quasi-concave. Let γ, δ ∈ R be so that 1 < γ < p and 1 < δ < q. Define F : R2 → R by F (u, v) = max{|u|γ , |v|δ }.

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Then F is convex, thus locally Lipschitz. Also, by Proposition 2.2, F is regular. Fix (u, v) ∈ R2 . If ξ1 ∈ ∂1 F (u, v), then |ξ1 | ≤ γ|u|γ−1, and if ξ2 ∈ ∂2 F (u, v), then |ξ2 | ≤ δ|v|δ−1 . It follows that F satisfies condition (C4). The relations

1 1 (−1, 0) + (1, 0) = 0 < 1 = F (−1, 0) = F (1, 0) 2 2

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F (0, 0) = F

show that F is not quasi-concave.

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