Báo cáo hóa học: "Research Article Existence of Solutions for a Class of Elliptic Systems in RN Involving the p x , q x -Laplacia" pptx

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 612938, 16 pages doi:10.1155/2008/612938 Research Article Existence of Solutions for a Class of Elliptic Systems in RN Involving the p x , q x -Laplacian S Ogras, R A Mashiyev, M Avci, and Z Yucedag Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey Correspondence should be addressed to R A Mashiyev, mrabil@dicle.edu.tr Received April 2008; Accepted 17 July 2008 Recommended by M Garcia Huidobro In view of variational approach, we discuss a nonlinear elliptic system involving the p x Laplacian Establishing the suitable conditions on the nonlinearity, we proved the existence of nontrivial solutions Copyright q 2008 S Ogras et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction The paper concerns the existence of nontrivial solutions for the following nonlinear elliptic system: −Δp x u ∂F x, u, υ ∂u in RN , −Δq x υ ∂F x, u, υ ∂υ in R , P,Q N where p x and q x are two functions such that < p x , q x < N N ≥ , for every x ∈ RN However, F ∈ C1 RN × R2 and Δp x is the p x -Laplacian operator defined by Δp x u div |∇u|p x −2 ∇u Using a variational approach, the authors prove the existence of nontrivial solutions Over the last decades, the variable exponent Lebesgue space Lp x and Sobolev space 1,p x 1–5 have been a subject of active research stimulated mainly by the development W of the studies of problems in elasticity, electrorheological fluids, image processing, flow in porous media, calculus of variations, and differential equations with p x -growth conditions 6–13 Among these problems, the study of p x -Laplacian problems via variational methods is an interesting topic A lot of researchers have devoted their work to this area 14–22 Journal of Inequalities and Applications The operator Δp x u : div |∇u|p x −2 ∇u is called p x -Laplacian, where p is a continuous nonconstant function In particular, if p x ≡ p constant , it is the well-known p-Laplacian operator However, the p x -Laplace operator possesses more complicated nonlinearity than p-Laplace operator due to the fact that Δp x is not homogeneous This fact implies some difficulties, as for example, we cannot use the Lagrange multiplier theorem and Morse theorem in a lot of problems involving this operator In literature, elliptic systems with standard and nonstandard growth conditions have been studied by many authors 23–28 , where the nonlinear function F have different and mixed growth conditions and assumptions in each paper In 29 , the authors show the existence of nontrivial solutions for the following pLaplacian problem: −Δp u ∂F x, u, υ ∂u in RN , −Δq υ ∂F x, u, υ ∂υ in RN , 1.1 where F ∈ C1 RN × R2 yields some mixed growth conditions and the primitive F being intimately connected with the first eigenvalue of an appropriate system Using a weak version of the Palais-Smale condition, that is, Cerami condition, they apply the mountain pass theorem to get the nontrivial solutions of the the system In 30 , the author obtains the existence and multiplicity of solutions for the following problem: − div |∇u|p x −2 ∇u ∂F x, u, υ ∂u in Ω, − div |∇υ|q x −2 ∇υ ∂F x, u, υ ∂υ in Ω, u 0, υ 1.2 on Ω, where Ω ⊂ RN is a bounded domain with a smooth boundary ∂Ω, N ≥ 2, p, q ∈ C Ω , p x > 1, q x > 1, for every x ∈ Ω The function F is assumed to be continuous in x ∈ Ω and of class C1 in u, υ ∈ R Introducing some natural growth hypotheses on the right-hand side of the system which will ensure the mountain pass geometry and PalaisSmale condition for the corresponding Euler-Lagrange functional of the system, the author limits himself to the subcritical case for function F to obtain the existence and multiplicity results In the paper 31 , Xu and An deal with the following problem: − div |∇u|p x −2 ∇u |u|p x −2 u ∂F x, u, υ ∂u in RN , − div |∇υ|q x −2 ∇υ |υ|q x −2 υ ∂F x, u, υ ∂υ in RN , 1.3 u, υ ∈ W 1,p x RN × W 1,q x RN , where N ≥ 2, p x , q x are functions on RN The function F is assumed to satisfy Caratheodory conditions and to be L∞ in x ∈ RN and C1 in u, υ ∈ R By the critical point S Ogras et al theory, the authors use the two basic results on the existence of solutions of the system; these results correspond to the sublinear and superlinear cases for p 2, respectively Inspired by the above-mentioned papers, we concern the existence of nontrivial solutions of problem P,Q We know that in the study of p x -Laplace equations in RN , the main difficulty arises from the lack of compactness So, establishing some growth conditions on the right-hand side of the system which will ensure the mountain pass geometry and Cerami condition for the corresponding Euler-Lagrange functional J and applying a subcritical case for function F, we will overcome this difficulty Notations and preliminaries 1,p x RN , We will investigate our problem P,Q in the variable exponent Sobolev space W0 p x N 1,p x R and W RN so we need to recall some theories and basic properties on spaces L Set C RN h ∈ C RN : inf h x > 2.1 x∈RN For every h ∈ C RN , denote h− : inf h x , h : x∈RN sup h x 2.2 x∈RN Let us define by U RN the set of all measurable real-valued functions defined on RN For p ∈ C RN , we denote the variable exponent Lebesgue space by Lp x RN u ∈ U RN : RN |u x | p x dx < ∞ , 2.3 which is equipped with the norm, so-called Luxemburg norm 1, 3, : |u|p x : |u|Lp x inf λ > : RN RN u x λ p x dx ≤ , 2.4 and Lp x RN , |·|Lp x RN becomes a Banach space, we call it generalized Lebesgue space Define the variable exponent Sobolev space W 1,p x RN by W 1,p x RN {u ∈ Lp x RN : |∇u| ∈ Lp x RN }, 2.5 and it can be equipped with the norm u 1,p x 1,p x The space W0 : u |u|p x W 1,p x |∇u|p x ∀u ∈ W 1,p x RN 2.6 ∞ RN is denoted by the closure of C0 RN in W 1,p x RN and it is 1,p x equipped with the norm for all u ∈ W0 u p x RN : |∇u|p x 1,p x ∀u ∈ W0 RN 1,p x If p− > 1, then the spaces Lp x RN , W 1,p x RN , and W0 reflexive Banach spaces 2.7 RN are separable and Journal of Inequalities and Applications Proposition 2.1 see 1, 3, The conjugate space of Lp x RN is Lp 1/p x For any u ∈ Lp x RN and v ∈ Lp x RN , we have RN uv dx ≤ p− p Proposition 2.2 see 1, 3, Denote p− p |u|p x , |u|p x ≤ |u|p x |v|p u p x − RN p x x x RN , where 1/p x ≤ 2|u|p x |v|p x 2.8 |u x | p x dx for all u ∈ Lp x RN , one has p− p u ≤ max |u|p x , |u|p x 2.9 Proposition 2.3 see Let p x and q x be measurable functions such that p x ∈ L∞ RN and ≤ p x q x ≤ ∞, for a.e x ∈ RN Let u ∈ Lq x RN , u / Then, p− p |u|p x q x ≤ ⇒ |u|p x q x ≤ |u|p x |u|p x q x ≥ ⇒ In particular, if p x p− |u|p x q x ≤ |u| ≤ |u|p x q x , q x p |u|p x q x p x ≤ 2.10 p is constant, then p ||u|p |q x |u|pq x Proposition 2.4 see 3, If u, un ∈ Lp x RN , n equivalent to each other: limn→∞ |un − u|p x limn→∞ un − u q x 2.11 1, 2, , then the following statements are 0, 0, un → u in measure in RN and limn→∞ un u Definition 2.5 < p x < N and for all x ∈ RN , let define p∗ x ⎧ ⎪ Np x ⎨ N−p x ⎪ ⎩ ∞ if p x < N, if p x ≥ N, where p∗ x is the so-called critical Sobolev exponent of p x Proposition 2.6 see 1, 32 Let p x ∈ C0,1 RN , that is, Lipschitz-continuous function defined on RN , then there exists a positive constant c such that |u|p∗ 1,p x for all u ∈ W0 x ≤c u p x , 2.12 RN In the following discussions, we will use the product space 1,p x Wp x ,q x : W0 1,q x RN × W0 RN , 2.13 S Ogras et al which is equipped with the norm u, υ p x ,q x : max u υ p x q x ∀ u, υ ∈ Wp x ,q x , 1,p x 2.14 1,q x where u p x resp., u q x is the norm of W0 RN resp., W0 RN The space ∗ Wp x ,q x denotes the dual space of Wp x ,q x and equipped with the norm · ∗,p x ,q x Thus, J u, υ where W −1,p 1,q x W0 x D1 J u, υ ∗,p x ,q x resp., W −1,q RN x ∗,p x D2 J u, υ ∗,q x , 2.15 1,p x RN is the dual space of W0 RN resp., R , and · ∗,p x resp., · ∗,q x is its norm For u, υ and ϕ, ψ in Wp x ,q x , let N F u, υ RN F x, u x , υ x dx 2.16 Then, F u, υ ϕ, ψ D1 F u, υ ϕ D2 F u, υ ψ , 2.17 where D1 F u, υ ϕ RN ∂F x, u, υ ϕ dx, ∂u 2.18 ∂F x, u, υ ψ dx RN ∂υ D2 F u, υ ψ The Euler-Lagrange functional associated to P,Q is defined by J u, υ |∇u|p x dx RN p x |∇υ|q x dx − u, υ RN q x 2.19 It is easy to verify that J ∈ C1 Wp x ,q x , R and that J u, υ ϕ, ψ D1 J u, υ ϕ D2 J u, υ ψ , 2.20 where D1 J u, υ ϕ D2 J u, υ ψ RN |∇u|p x −2 ∇u∇ϕ dx − D1 F u, υ ϕ , 2.21 q x −2 RN |∇υ| ∇υ∇ψ dx − D2 F u, υ ψ Definition 2.7 u, υ is called a weak solution of the system P,Q if RN |∇u|p x −2 ∇u∇ϕ dx for all ϕ, ψ ∈ Wp x ,q x RN |∇υ|q x −2 ∇υ∇ψ dx ∂F x, u, υ ϕ dx RN ∂u ∂F x, u, υ ψ dx, RN ∂υ 2.22 Journal of Inequalities and Applications Definition 2.8 We say that J satisfies the Cerami condition C if every sequence ωn Wp x ,q x such that |J ωn | ≤ c, ωn J ωn −→ ∈ 2.23 contains a convergent subsequence in the norm of Wp x ,q x In this paper, we will use the following assumptions: F1 F ∈ C1 RN × R2 , R and F x, 0, 0; F2 for all u, υ ∈ R and for a.e x ∈ R N ∂F x, u, υ ∂u ≤ a1 x | u, υ |p ∂F x, u, υ ∂υ − −1 a2 x | u, υ |p −1 , 2.24 q− −1 q −1 ≤ b1 x | u, υ | b2 x | u, υ | , where < p− , ∈ Lδ δ x q x x q < p∗ − , q∗ − , q− ≤ p , RN ∩ Lβ x RN , p x , p x −1 bi ∈ Lγ q x , q x −1 γ x q∗ x q x , x −q x β x q∗ RN ∩ Lβ x RN , x p∗ x p x q∗ i 1, 2, p∗ x p x , p∗ x − p x p∗ x q∗ x x − p∗ x q∗ x 2.25 ; u, υ ·∇F x, u, υ − F x, u, υ ≤ for all x, u, υ ∈ RN × R2 \ { 0, }, where ∇F ∂F/∂u, ∂F/∂υ ; F3 F4 suppose there exist two positive and bounded functions a ∈ LN/p x RN and b ∈ LN/q x RN such that lim sup | u,υ |→0 p x q x |F x, u, υ | q x a x |u|p x < λ1 < lim | u,υ |→ ∞ inf p x b x |υ|q x p x q x |F x, u, υ | q x a x |u|p x p x b x |υ|q x 2.26 Let λ1 denote the first eigenvalue of the nonlinear eigenvalue problem in RN : −Δp x u λa x |u|p x −2 u in RN , −Δq x υ λb x |υ|q x −2 υ in RN 2.27 It is useful to recall the variational characterization: λ1 inf RN RN 1/p x |∇u|p x a x /p x |u|p x 1/q x |∇υ|q x dx b x /q x |υ|q x dx : u, υ ∈ Wp x ,q x \ { 0, } 2.28 We will assume that λ1 is a positive real number for all u, υ ∈ Wp x ,q x \ { 0, } For more details about the eigenvalue problems, we refer the reader to 17 S Ogras et al The main results We will use the mountain pass theorem together with the following lemmas to get our main results Lemma 3.1 Under the assumptions F1 and F2 , the functional F is well defined, and it is of class C1 on Wp x ,q x Moreover, its derivative is ∂F x, u, υ ω dx RN ∂u F u, υ ω, z ∂F x, u, υ z dx RN ∂υ ∀ u, υ , ω, z ∈ Wp x ,q x 3.1 Proof For all pair of real functions u, υ ∈ Wp x ,q x , under the assumptions F1 and F2 , we can write u F x, u, υ u ∂F x, s, υ ds ∂s F x, u, υ ≤ c1 a1 x |u|p − F x, 0, υ |υ|p − −1 a2 x |u|p |u| υ ∂F x, s, υ ds ∂s |υ|p ∂F x, 0, s ds ∂s −1 b1 x |υ|q |u| − F x, 0, , b2 x |υ|q 3.2 Then, RN F x, u, υ dx ≤ c2 − RN a1 x |u|p dx RN p −1 RN a2 x |υ| a1 x |υ|p − −1 |u|dx RN a2 x |u|p dx 3.3 q− |u|dx RN q b1 x |υ| dx RN b2 x |υ| dx , if we consider the fact that 1,p x W0 RN → Lp − p x p− − ⇒ ||u|p |p x RN p− p x |u|p− p x ≤ c u and if we apply Propositions 2.1, 2.3, and 2.6 and take ∈ Lδ Lγ x RN , then we have RN F x, u, υ dx ≤ 2c1 |a1 |δ − x ||u|p |p x |a2 |β x ||υ|p 2c1 |a1 |δ |a1 |β x ||υ|p −1 |q ∗ p− x x x u p− p x |a2 |β x υ −1 |u|p∗ |b1 |γ x x x ||υ|q |q x |a1 |β x |υ| p− −1 q∗ |u|p∗ |a1 |β x υ p −1 q x u p x x |a2 |δ x − x |u|p∗ q− |b1 |γ x |υ|q− q x p− −1 q x u p x |b1 |γ x υ ||u|p |p x x ||υ|q |q x p x |u|p p x q |b2 |γ x |υ|q q x x u p p x |b2 |γ x υ |a2 |δ q− q x x |b2 |γ |a2 |δ x 3.4 RN ∩ Lβ x RN , bi ∈ x |q ∗ p− −1 |u|p− p x p −1 |a2 |β x |υ| p −1 q∗ x ≤ c3 |a1 |δ |u|p∗ − for p− > 1, q q x < ∞ 3.5 Journal of Inequalities and Applications Hence, F is well defined Moreover, one can see easily that F is also well defined on Wp x ,q x Indeed, using F2 for all ω, z ∈ Wp x ,q x , we have ∂F x, u, υ ω dx RN ∂u F u, υ ω, z F u, υ ω, z ≤ a1 x | u, υ |p RN RN a1 x |u|p RN RN RN −1 b1 x | u, υ |q RN ≤ − − −1 a2 x |u|p b1 x |u|q − b2 x |u|q ∂F x, u, υ z dx, RN ∂υ − −1 −1 −1 RN |ω|dx a1 x |υ|p RN |z|dx RN |z|dx RN − −1 a2 x |υ|p b1 x |υ|q − b2 x |υ|q |ω|dx −1 b2 x | u, υ |q |ω|dx −1 −1 a2 x | u, υ |p |z|dx |ω|dx −1 3.6 |ω|dx −1 |z|dx −1 |z|dx, and applying Propositions 2.1, 2.3, and 2.6 and considering the conditions p x > p x and q x > q x , it follows that ∂F x, u, υ ω dx ≤ |a1 |δ RN ∂u |a2 |δ ≤ |a1 |δ x − −1 |p ∗ ||u|p −1 ||u|p x x |p ∗ − p −1 |u| p− −1 p∗ x |a2 |δ ≤ c4 |a1 |δ x x x |ω|p x x p −1 −1 p∗ x |u| p u p− −1 p x −1 |ω|p∗ x |q ∗ − p −1 |a1 |β x |υ| p− −1 q∗ x x |a2 |β x |υ| p p− −1 q x |a2 |δ u x x |ω|p∗ |ω|p∗ p −1 −1 q∗ x |ω|p x |a1 |β x υ |q ∗ |a2 |β x ||υ|p |ω|p x |ω|p x − −1 |a1 |β x ||υ|p x |ω|p∗ p −1 p x x x |a2 |β υ p −1 q x ω p x < ∞, 3.7 and similarly ∂F x, u, υ z dx RN ∂υ ≤ c5 |b1 |β x u q− −1 p x |b1 |γ x υ q− −1 q x 3.8 |b2 |β x u q −1 p x |b2 |γ x υ q −1 q x z q x < ∞ Now let us show that F is differentiable in sense of Fr´ chet, that is, for fixed u, υ ∈ e Wp x ,q x and given ε > 0, there must be a δ ε, u, υ > such that |F u ω, υ z − F u, υ − F u, υ ω, z | ≤ ε ω for all ω, z ∈ Wp x ,q x with ω p x z q x ≤ δ p x z q x , 3.9 S Ogras et al Let Br be the ball of radius r which is centered at the origin of RN and denote Br 1,p x 1,q x Br × W Br as follows: RN − Br Moreover, let us define the functional Fr on W0 Fr u, υ F x, u x , υ x dx 3.10 Br 1,p x If we consider F1 and F2 , it is easy to see that Fr ∈ C1 W0 1,p x 1,q x addition for all ω, z ∈ W0 Br × W Br , and in Br , we have ∂F x, u, υ ω dx Br ∂u Fr u, υ ω, z 1,q x Br × W ∂F x, u, υ z dx Br ∂υ 3.11 ∗ Also as we know, the operator Fr : Wp x ,q x → Wp x ,q x is compact Then, for all u, υ , ω, z ∈ Wp x ,q x , we can write F u z − F u, υ − F u, υ ω, z ω, υ ≤ Fr u z − Fr u, υ − Fr u, υ ω, z ω, υ F x, u 3.12 ∂F ∂F x, u, υ ω − x, u, υ z dx z − F x, u, υ − ∂u ∂υ ω, υ Br By virtue of the mean-value theorem, there exist ζ1 , ζ2 ∈ 0, such that F x, u ∂F x, u ∂u z − F x, u, υ ω, υ ∂F x, u, υ ∂υ ζ1 ω, υ ω ζ2 z z 3.13 Using the condition F2 , we have Br ∂F x, u ∂u ≤ ∂F x, u, υ ∂υ ζ1 ω, υ ω ζ1 ω|p a1 x |u − −1 − |u|p − ζ2 z z − −1 ∂F ∂F x, u, υ ω − x, u, υ z dx ∂u ∂υ −1 ζ1 ω|p a2 x |u −1 − |u|p |ω|dx 3.14 Br ζ2 z|q b1 x |υ − −1 − |υ|q − −1 ζ2 z|q b2 x |υ −1 − |υ|q −1 |z|dx Br By help of the elementary inequality |a ≤ 2p − −1 a1 x |u|p −1 − −1 b|s ≤ 2s−1 |a|s |ω|dx ζ1 p− −1 Br 2p −1 |b|s for a, b ∈ RN , we can write a1 x |ω|p a2 x |u|p −1 −1 |ω|dx ζ1 p −1 −1 |ω|dx q− −1 −1 b1 x |υ| Br −1 |ω|dx |z|dx ζ2 3.15 q− −1 q− −1 b1 x |z| |z|dx Br b2 x |υ|q −1 a2 x |ω|p Br Br 2q −1 Br Br q− −1 − −1 |z|dx ζ2 q −1 b2 x |z|q Br −1 |z|dx, 10 Journal of Inequalities and Applications applying Propositions 2.1, 2.3, and 2.6, then we have ≤ c6 |a1 |δ ||u|p x − −1 |p ∗ −1 |a2 |δ ||u|p |b1 |γ x ||υ|q |b2 |γ ≤ c7 x x ||υ|q |a1 |δ x |b1 |γ u x x |ω|p x |a1 |δ |p ∗ ||ω|p −1 |p∗ x |ω|p x −1 |q∗ x |z|q x |b1 |γ x ||z|q −1 − |q∗ x |z|q x |b2 |γ x ||z|q p− −1 p x |a1 |δ q− −1 q x υ ω x |b1 |γ x |a2 |δ − −1 ||ω|p x x p− −1 p x z − |ω|p x |p∗ x |ω|p x −1 |q ∗ x |z|q x −1 |q ∗ x |z|q x |a2 |δ q− −1 q x x u x |b2 |γ υ x 3.16 , p −1 p x |a2 |δ q −1 q x x |b2 |γ x ω p −1 p x ω p x z q −1 q x z q x , and by the fact that |ai |Lδ x −→ 0, |bi |Lγ x for i Br Br −→ 3.17 1, 2, as r → ∞, and for r sufficiently large, it follows that F x, u ω, υ z − F x, u, υ − Br ∂F ∂F x, u, υ ω − x, u, υ z dx ≤ ε ω ∂u ∂υ p x z q x 3.18 It remains only to show that F is continuous on Wp x ,q x Let un , υn , u, υ Wp x ,q x such that un , υn → u, υ Then, for ω, z ∈ Wp x ,q x , we have ∈ |F un , υn ω, z − F u, υ ω, z | ≤ |Fr un , υn ω, z − Fr u, υ ω, z | ∂F x, u, υ ∂u ω dx Br ∂F x, un , υn ∂u ∂F x, u, υ ∂υ z dx , Br ∂F x, un , υn ∂υ 3.19 then by F2 , we can write a1 x |un |p − −1 |u|p − −1 |υn |p − −1 |υ|p − −1 |ω|dx 3.20 Br a2 x |un |p −1 |u|p −1 |u|q −1 |u|q −1 |υn |p −1 |υn |q −1 |υn |q −1 |υ|p −1 |υ|q −1 |υ|q −1 |ω|dx I1 −1 |z|dx I2 −1 |z|dx 3.21 Br b1 x |un |q − − − − Br b2 x |un |q Br S Ogras et al 11 Thus, a1 x |un |p I1 ≤ − −1 a1 x |u|p |ω|dx Br − −1 a1 x |υn |p |ω|dx Br a1 x |υ|p − −1 a2 x |un |p −1 −1 |ω|dx −1 |ω|dx Br a2 x |υ|p |ω|dx Br a2 x |u|p |ω|dx Br a2 x |υn |p −1 Br |ω|dx Br − −1 3.22 |ω|dx Br ≤ c9 |a1 |δ x |a2 |δ p− −1 un p x x un x u p− −1 p x |a2 |δ x u |a1 |δ p −1 p x |a1 |β x υn p −1 p x p− −1 q x |a2 |β x υn |a1 |β x υ p −1 q x p− −1 q x |a2 |β x υ p −1 q x ω q −1 q x z p x Similarly, I2 ≤ c10 |b1 |β x un q− −1 p x |b2 |β x un |b1 |β x u q −1 p x q− −1 p x |b2 |β x u 1,p x Since Fr is continuous on W0 |b1 |γ x υn q −1 p x |b2 |γ 1,q x Br × W x q− −1 q x υn |b1 |γ x υ q −1 q x |b2 |γ x q− −1 q x υ 3.23 q x Br , then we have |Fr un , υn ω, z − Fr u, υ ω, z | −→ 0, 3.24 as n → ∞ Moreover, using 3.17 , when r sufficiently large, I1 and I2 tend also to Hence, |F un , υn ω, z − F u, υ ω, z | −→ 0, 3.25 as un , υn → u, υ , this implies F is continuous on Wp x ,q x ∗ Lemma 3.2 Under the assumptions F1 and F2 , F is compact from Wp x ,q x to Wp x ,q x Proof Let un , υn be a bounded sequence in Wp x ,q x Then, there exists a subsequence we denote again as un , υn which converges weakly in Wp x ,q x to a u, υ ∈ Wp x ,q x Then, if we use the same arguments as above, we have |F un , υn ω, z − F u, υ ω, z | ≤ |Fr un , υn ω, z − Fr u, υ ω, z | ω dx Br ∂F ∂F x, un , υn − x, u, υ ∂u ∂u z dx Br ∂F ∂F x, un , υn − x, u, υ ∂υ ∂υ 1,p x 3.26 1,q x Since the restriction operator is continuous, then un , υn u, υ in W0 Br × W Br Because of the compactness of Fr , the first expression on the right-hand side of the equation tends to 0, as n → ∞, and as we did above, when r sufficiently large, I1 and I2 tend also to ∗ This implies F is compact from Wp x ,q x to Wp x ,q x 12 Journal of Inequalities and Applications Lemma 3.3 If F1 , F2 , and F3 hold, then J satisfies the condition C , that is, there exists a sequence un , υn ∈ Wp x ,q x such that i |J un , υn | ≤ c, ii un υn p x J un , υn q x → as n → ∗,p x ,q x ∞ contains a convergent subsequence Proof By the assumption ii , it is clear that J un , υn ω, z ≤ ξn → as n → ∞ for all un , υn , then we have ω, z ∈ Wp x ,q x Let us choose ω, z ξn ≥ J un , υn un , υn ≥ un p− p x υn q− q x − ∂F x, un , υn υn dx ∂υ ∂F x, un , υn un ∂u RN 3.27 F x, un , υn dx 3.28 Moreover, by the assumption i , we can write c ≥ −J un , υn ≥ − un p p− p x − υn q q− q x RN Using the assumption F3 , it follows that ξn c ≥ J un , υn un , υn − J un , υn ≥ 1− − ≥ p p− p x 1− ∂F x, un , υn un ∂u RN 1− un p un p− p x 1− q υn q− q x RN F x, un , υn dx 3.29 ∂F x, un , υn υn dx ∂υ q υn q− q x Thus, the sequence un , υn is bounded in Wp x ,q x Then, there exists a subsequence we denote again as un , υn which converges weakly in Wp x ,q x We recall the elementary inequalities: 22−p |a − b|p ≤ a|a|p−2 − b|b|p−2 · a − b p − |a − b|2 |a| |b| p−2 if p ≥ 2, ≤ a|a|p−2 − b|b|p−2 · a − b if < p < 2, 3.30 3.31 for all a, b ∈ RN , where · denotes the standard inner product in RN We will show that un , υn contains a Cauchy subsequence Let us define the sets Up {x ∈ RN : p x ≥ 2}, Vp {x ∈ RN : < p x < 2}, Uq {x ∈ RN : q x ≥ 2}, Vq {x ∈ RN : < q x < 2} 3.32 For all x ∈ RN , we put Φn,k |∇un |p x −2 ∇un − |∇uk |p x −2 ∇uk · ∇un − ∇uk , 3.33 Ψn,k |∇un | |∇uk | 2−p x S Ogras et al 13 Therefore for p x ≥ 2, using 3.30 , we have 22−p Up |∇un − ∇uk |p x dx ≤ |∇un |p x −2 ∇un − |∇uk |p x −2 ∇uk · ∇un − ∇uk dx Up ≤ Φn,k dx : Tn,k RN D1 J un , υn −D1 J uk , υk ≤ D1 J un , υn D1 F un , υn −D1 F uk , υk D1 J uk , υk ∗,p x ,q x D1 F un , υn − D1 F uk , υk un − uk ∗,p x ,q x ∗,p x ,q x un − uk un −uk p x p x 3.34 When < p x < 2, employing 3.31 and Proposition 2.2, it follows |∇un −∇uk |p x dx ≤ Vp |∇un −∇uk |p x |∇un | |∇uk | p x p x −2 /2 |∇un | |∇uk | Vp −p x /2 RN 2/p x |∇un −∇uk |2 Ψ−1 dx n,k × Ψn,k p− /2 2/ 2−p x |∇un −∇uk |2 Ψ−1 dx n,k , p x / 2−p x RN Ψn,k p− − ≤ max × max RN −p− /2 2−p /2 dx p− /2 p x / 2−p x Ψn,k p x / 2−p x , · Tn,k , p− − RN −p /2 Ψn,k p /2 p x / 2−p x , 2−p /2 dx · Tn,k 2−p− /2 dx p /2 RN − × max dx p x /2 ≤ |∇un − ∇uk |p x Ψn,k ≤ max p x 2−p x /2 RN Ψn,k 2−p /2 dx 3.35 1,p x Since Tn,k is uniformly bounded in W0 RN in accordance with n, k, and by the fact that J um , υm ∗,p x ,q x → as m → ∞, F is compact and by Proposition 2.4, we have lim n,k→ ∞ RN |∇un − ∇uk |p x dx 3.36 Applying the same arguments, we can find a subsequence of un , υn such that lim n,k→ ∞ RN |∇υn − ∇υk |q x dx 3.37 Therefore by Proposition 2.2, for a convenient subsequence, we have lim n,k→ ∞ un , υn − uk , υk ∗,p x ,q x 3.38 Hence, un , υn contains a Cauchy subsequence and so contains a strongly convergent subsequence The proof is complete 14 Journal of Inequalities and Applications Lemma 3.4 Under the assumptions F1 – F4 , the functional J satisfies the following i There exists ρ, σ > such that u υ p x ii There exists E ∈ Wp x ,q x such that E Proof By F4 , we can find ρ > such that RN p x ,q x > ρ and J E ≤ υ p x a x |u|p x p x F x, u, υ < λ1 F x, u, υ < λ1 u ρ implies J u, υ ≥ σ > q x b x |υ|q x q x , 3.39 b x |υ|q x q x dx, |∇υ|q x q x dx, a x |u|p x p x RN ρ, so we have q x since λ1 > 0, then we have RN 0< RN F x, u, υ < |∇u|p x p x RN |∇u|p x p x |∇υ|q x q x dx − RN F x, u, υ 3.40 J u, υ Hence, there exists σ > such that J ≥ σ > Let τ, θ be an eigenfunction relative to λ1 Then, using the assumption F4 , we can obtain for > and t sufficiently large, F x, t1/p x τ, t1/q x θ ≥ t λ1 a x |τ|p x p x b x |θ|q x q x 3.41 Thus, J t1/p x τ, t1/q x θ t − ≤ |∇τ|p x p x RN RN RN |∇θ|q x q x dx F x, t1/p x τ, t1/q x θ dx |∇τ|p x p x − t λ1 |∇θ|q x q x a x |τ|p x dx RN p x 3.42 dx b x |θ|q x dx , RN q x then it follows J t1/p x τ, t1/q x θ ≤ − t p RN a x |τ|p x dx q RN b x |θ|q x dx 3.43 −∞ Hence, for t sufficiently large, So, we can conclude that limt→ ∞ J t1/p x τ, t1/q x θ J t1/p x τ, t1/q x θ ≤ As a consequence, we can say that the functional J u, υ has a critical point; and as we know, the critical points of J u, υ are the weak solutions of the system P,Q Theorem 3.5 The system P,Q has at least 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,” Nonlinear Analysis: Theory, Methods & Applications, vol 68, no 4, pp 956–968, 2008 32 L Diening, “Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp · and W k,p · ,” Mathematische Nachrichten, vol 268, no 1, pp 31–43, 2004  ... ∗ ,p x ,q x Thus, J u, υ where W −1 ,p 1 ,q x W0 x D1 J u, υ ∗ ,p x ,q x resp ., W −1 ,q RN x ∗ ,p x D2 J u, υ ∗ ,q x , 2.15 1 ,p x RN is the dual space of W0 RN resp ., R , and · ∗ ,p x resp ., · ∗ ,q x. .. ≤ ? ?, for a. e x ∈ RN Let u ∈ Lq x RN , u / Then, p? ?? p |u |p x q x ≤ ⇒ |u |p x q x ≤ |u |p x |u |p x q x ≥ ⇒ In particular, if p x p? ?? |u |p x q x ≤ |u| ≤ |u |p x q x , q x p |u |p x q x p x ≤ 2.10 p. .. b1 x | u, υ | b2 x | u, υ | , where < p? ?? , ∈ Lδ δ x q x x q < p? ?? − , q? ?? − , q? ?? ≤ p , RN ∩ Lβ x RN , p x , p x −1 bi ∈ Lγ q x , q x −1 γ x q? ?? x q x , x ? ?q x β x q? ?? RN ∩ Lβ x RN , x p? ?? x p x q? ??