Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 191649, 12 pages doi:10.1155/2009/191649 Research Article Multiple Solutions for a Class of p x -Laplacian Systems Yongqiang Fu and Xia Zhang Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Correspondence should be addressed to Xia Zhang, piecesummer1984@163.com Received 19 November 2008; Accepted 11 February 2009 Recommended by Ondrej Dosly We study the multiplicity of solutions for a class of Hamiltonian systems with the p x -Laplacian Under suitable assumptions, we obtain a sequence of solutions associated with a sequence of positive energies going toward infinity Copyright q 2009 Y Fu and X Zhang 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 and Main Results Since the space Lp x and W 1,p x were thoroughly studied by Kov´ cik and R´ kosn´k , aˇ a ı variable exponent Sobolev spaces have been used in the last decades to model various phenomena In , Ruˇ iˇ ka presented the mathematical theory for the application of variable ˚z c exponent spaces in electro-rheological fluids In recent years, the differential equations and variational problems with p x -growth conditions have been studied extensively; see for example 3–6 In , De Figueiredo and Ding discussed the multiple solutions for a kind of elliptic systems on a smooth bounded domain Motivated by their work, we will consider the following sort of p x -Laplacian systems with “concave and convex nonlinearity”: −div ∇u −div ∇v p x −2 p x −2 ∇u ∇v u x |u|p x −2 u |v|p x −2 v v x 0, Hu x, u, v , −Hv x, u, v , x ∈ Ω, x ∈ Ω, 1.1 x ∈ ∂Ω, where Ω ⊂ RN is a bounded domain, p is continuous on Ω and satisfies < p− ≤ p x ≤ p < N, and H : Ω × R2 → R is a C1 function In this paper, we are mainly interested in the class Journal of Inequalities and Applications of Hamiltonians H such that where < α− ≤ α x ≤ p x , p x p |v|β x β x |u|α x α x H x, u, v F x, u, v , 1.2 p∗ x Here we denote β x sup p x , p− x∈Ω inf p x , 1.3 x∈Ω and denote by p x β x the fact that infx∈Ω β x − p x > Throughout this paper, F x, u, v satisfies the following conditions: H1 F ∈ C1 Ω × R2 , R Writing z u, v , F x, ≡ 0, Fz x, ≡ 0; p∗ x , < q2− ≤ q2 x < p x such that H2 there exist p x < q1 x Fu x, u, v , Fv x, u, v ≤ a0 |u|q1 where a0 is positive constant; H3 there exist μ x , ν x ∈ C1 Ω with p x and R0 > such that Fu x, u, v u μ x x −1 |v|q2 x −1 , 1.4 p∗ x , < ν− ≤ ν x ≤ p x , μ x Fv x, u, v v ≥ F x, u, v > 0, ν x 1.5 when | u, v | ≥ R0 As 8, Lemma 1.1 , from assumption H3 , there exist b0 , b1 > such that F x, u, v ≥ b0 |u|μ x |v|ν − b1 , x 1.6 for any x, u, v ∈ Ω × R2 We can also get that there exists b2 > such that Fu x, u, v u μ x Fv x, u, v v ν x b2 ≥ F x, u, v , 1.7 for any x, u, v ∈ Ω × R2 In this paper, we will prove the following result Theorem 1.1 Assume that hypotheses (H1)–(H3) are fulfilled If F x, z is even in z, then problem 1.1 has a sequence of solutions {zn } such that I zn as n → ∞ ∇un Ω p x p x un p x − ∇vn p x p x p x − H x, zn dx −→ ∞, 1.8 Journal of Inequalities and Applications Preliminaries First we recall some basic properties of variable exponent spaces Lp x Ω and variable exponent Sobolev spaces W 1,p x Ω , where Ω ⊂ RN is a domain For a deeper treatment on these spaces, we refer to 1, 9–11 Let P Ω be the set of all Lebesgue measurable functions p : Ω → 1, ∞ and |u|p x inf λ > : Ω u λ p x dx ≤ 2.1 The variable exponent space Lp x Ω is the class of all functions u such that Ω |u x |p x dx < ∞ Under the assumption that p < ∞, Lp x Ω is a Banach space equipped with the norm 2.1 The variable exponent Sobolev space W 1,p x Ω is the class of all functions u ∈ p x Ω such that |∇u| ∈ Lp x Ω and it can be equipped with the norm L u |u|p x 1,p x |∇u|p x 2.2 For u ∈ W 1,p x Ω , if we define |||u||| then |||u||| and u 1,p x By W0 inf λ > : |u|p x |∇u|p x λp x Ω dx ≤ , 2.3 are equivalent norms on W 1,p x Ω 1,p x ∞ Ω we denote the subspace of W 1,p x Ω which is the closure of C0 Ω 1,p x with respect to the norm 2.2 and denote the dual space of W0 Ω by W −1,p x Ω We N know that if Ω ⊂ R is a bounded domain, ||u||1,p x and |∇u|p x are equivalent norms on 1,p x W0 Ω 1,p x Under the condition < p− ≤ p < ∞, W0 space, then there exist {en }n∞ ⊂ 1,p x W0 Ω and 1,p x W −1,p x x Ω span ei : i Ω such that 1, , n, , Ω span fj : j 2.4 1, , m, In the following, we will denote that E E1 ⊂ W −1,p if n m, if n / m, fm en W0 Ω is a separable and reflexive Banach {fm }m∞1 1,p x {0} × W0 Ω , E1 ⊕ E2 , where E2 1,p x W0 Ω × {0} 2.5 Journal of Inequalities and Applications For any z ∈ E, define the norm ||z|| en , and 0, en , en || u, v || 1 span e1 , , en ⊕ E2 , Xn denote the complement of X n in E by X n ⊥ |||u||| |||v||| For any n ∈ N, set en 2 E1 ⊕ span e1 , , en , Xn 2.6 2 span{en , en , } The Proof of Theorem 1.1 Definition 3.1 We say that z0 Ω ∇u0 − v0 p x −2 u0 , v0 ∈ E is a weak solution of problem 1.1 , that is, ∇u0 ∇u p x −2 u0 p x −2 u0 u − ∇v0 p x −2 ∇v0 ∇v 3.1 v0 v − Hu x, u0 , v0 u − Hv x, u0 , v0 v dx In this section, we denote that Vm positive constant, for any i 0, 1, ∀z ∈ E 0, 1, , m}, for any m ∈ N, and ci is span{ei : i Lemma 3.2 Any PS sequence {zn } ⊂ E, that is, |I zn | ≤ c and I zn bounded → 0, as n → ∞, is Proof Let s > be sufficiently small such that l1 infx∈Ω 1/p x − s /μ x > 0, l2 infx∈Ω s /ν x − 1/p x > 0, l3 supx∈Ω 1/α x − s /μ x > 0, l4 supx∈Ω s /ν x − 1/ β x > Let {zn } ⊂ E be such that |I zn | ≤ c and I zn → 0, as n → ∞ We get s s un , μ x ν x I zn − I zn , Ω s − p x μ x ∇un 1 s − ν x p x 1 s − μ x α x Ω l1 ∇un p x s un μ x un l2 ∇vn ∇un p x p x −2 un p x ∇vn s Fu x, un , un μ x ≥ p x p x s un μ x p x − ∇un s ν x p x −2 ∇vn ∇un ∇μ p x −2 ∇vn ∇ν s Fv x, un , vn − F x, un , ν x αx 1 s − ν x β x sF x, un , − l3 un ∇un ∇μ − s ν x αx ∇vn β x dx l4 p x −2 β x ∇vn ∇ν − s b2 dx 3.2 Journal of Inequalities and Applications As μ x , ν x ∈ C1 Ω , by the Young inequality, we can get that for any ε1 , ε2 ∈ 0, , s un μ x ∇un p x −2 ∇un ∇μ ≤ c0 ∇un ≤ c0 p x −1 s ν x ∇vn p x −2 1−p x ε1 p x − p x ≤ c0 ε1 ∇un un p x ∇un 1−p ε1 ∇vn ∇ν ≤ c1 ε2 ∇vn ε1 p x p x un p x p x 1−p ε2 un p x 3.3 , p x Let ε1 , ε2 be sufficiently small such that c0 ε1 ≤ l1 , c1 ε2 ≤ l2 , 3.4 then I zn − I zn , ≥ Ω s s un , μ x ν x l1 ∇un l2 ∇vn p x − l3 un p x 1−p αx c0 ε1 un s b0 un p x μ x l4 b0 β x ν x − b1 1−p − c1 ε2 p x − s b2 dx 3.5 Note that α x ≤ p x we get un αx ε3 α x un ≤ μ x ≤ ε3 un un p x ≤ p x ≤ μ x ε4 p x μ x ≤ ε4 un β x , by the Young inequality, for any ε3 , ε4 , ε5 ∈ 0, , μ x ,p x μ x ε5 p x β x ≤ ε5 un β x μ x μ x − α x α x / α x −μ x ε3 μ x −α / μ−α ε3 μ x − , β x −p x p x / ε5 β x −p / β−p ε5 , μ x − p x p x / p x −μ x ε4 μ x −p / μ−p ε4 β x − − p x −β x 3.6 Journal of Inequalities and Applications 1−p Let ε3 , ε4 , ε5 be sufficiently small such that l3 ε3 get I zn − I zn , s s un , μ x ν x ≥ c0 ε1 1−p ε4 ≤ sb0 and c1 ε2 l1 ∇un Ω l2 ∇vn p x p x ε5 ≤ l4 , then we − c2 dx 3.7 Note that I zn , s s un , μ x ν x ≤ ≤ c3 I zn ≤ c4 I zn s ν x s un μ x · I zn s un μ x ∇ · · ∇un ∇ p x ∇vn p x p x s ν x p x , 3.8 and for n ∈ N being large enough, we have ≤ c4 I zn l1 l2 , 4 3.9 It is easy to know that if |∇un |p x ≥ and |∇vn |p x ≥ 1, ∇un p x ≤ Ω ∇un p x dx, ∇vn p x ≤ Ω ∇vn p x dx, 3.10 thus we get I zn ≥ Ω l1 ∇un p x l2 ∇vn p x − c2 dx, 3.11 then |∇un |p x , |∇vn |p x are bounded Similarly, if |∇un |p x < or |∇vn |p x < 1, we can also get that |∇un |p x , |∇vn |p x are bounded It is immediate to get that {zn } is bounded in E Lemma 3.3 Any PS sequence contains a convergent subsequence Proof Let {zn } ⊂ E be a PS sequence By Lemma 3.2, we obtain that {zn } is bounded in E As E is reflexive, passing to a subsequence, still denoted by {zn }, we may assume that there Journal of Inequalities and Applications 1,p x exists z ∈ E such that zn → z weakly in E Then we can get un → u weakly in W0 Note that I zn − I z , un − u, ∇un Ω un − un p x −2 ∇un − ∇u p x −2 α x −2 p x −2 un − |u|p x −2 u un − |u| α x −2 Ω ∇u ∇ un − u un − u 3.12 un − u u − Fu x, un , − Fu x, u, v un − u dx It is easy to get that I zn − I z , un − u, Ω −→ 0, 3.13 Fu x, u, v un − u dx −→ 0, and un → u in Lp x Ω , un → u in Lα x Ω , as n → ∞ Then Ω Ω un p x −2 un − |u|p x −2 u un − u dx −→ 0, 3.14 un α x −2 un − |u| α x −2 un − u dx −→ 0, u as n → ∞ By condition H2 , we obtain Ω Fu x, un , ≤ Ω ≤ a1 un − u dx a0 un − u un q1 x −1 un q2 x −1 q1 x −1 q1 x un − u dx · un − u Ω ∇un Fu x, un , p x −2 q1 x It is immediate to get that |un − u|1 → 0, ||un |q1 |un − u|q1 x → 0, |un − u|q2 x → 0, then we get Ω 3.15 x −1 | q1 x q2 x −1 q2 x , ||vn |q2 x −1 · un − u | q2 x q2 x are bounded and un − u dx −→ 0, 3.16 ∇un − ∇u p x −2 ∇u ∇ un − u dx −→ 0, as n → ∞ Similar to 3, 4, Theorem 3.1 , we divide Ω into two parts: Ω1 {x ∈ Ω : p x < 2}, Ω2 {x ∈ Ω : p x ≥ 2} 3.17 Journal of Inequalities and Applications On Ω1 , we have Ω1 ∇un − ∇u ≤ c5 p x dx ∇un Ω1 × ≤ c6 then Ω1 Ω2 ∇un ∇un ∇un × p x −2 p x p x −2 p x p x −2 ∇un − ∇u ∇un − ∇u ∇u p x ∇un − ∇u 2−p x /2 p x ∇u ∇u p x −2 ∇u p x /2 3.18 dx ∇un − ∇u 2−p x /2 2/ 2−p x ,Ω1 p x /2 2/ p x ,Ω1 , |∇un − ∇u|p x dx → On Ω2 , we have |∇un − ∇u|p x dx ≤ c7 Thus we get → v in Ω2 |∇un |p x −2 ∇un − |∇u|p x −2 ∇u ∇un − ∇u dx −→ 1,p x |∇un − ∇u|p x dx → Then un → u in W0 Ω 1,p x W0 3.19 Ω , as n → ∞ Similarly, Ω Lemma 3.4 There exists Rm > such that I z ≤ for all z ∈ X m with ||z|| ≥ Rm u, v ∈ X m , u ∈ Vm , we have Proof For any z I z ≤ ∇u p x |u|p x − p x Ω ∇v p x |v|p x p x − F x, u, v dx 3.20 ≤ ∇u p x |u| p x − p− Ω In the following, we will consider i If |||u||| ≤ We have ∇u p x p− Ω ∇v Ω |u|p x p x p |∇u|p x sup p y < μx y∈Q x − b0 |u|μ x b1 dx |u|p x /p− − b0 |u|μ x dx − b0 |u|μ x ii If |||u||| > Note that μ, p ∈ C Ω , p x which is an open subset of Ω such that px |v| p x dx ≤ p− 3.21 μ x For any x ∈ Ω, there exists Q x inf μ y , y∈Q x 3.22 Journal of Inequalities and Applications then {Q x }x∈Ω is an open covering of Ω As Ω is compact, we can pick a finite subcovering n {Q x }n for Ω Thus there exists a sequence of open set {Ωi }n such that Ω i i i Ωi and pi sup p x < μi− inf μ x , for i 3.23 x∈Ωi x∈Ωi |||u|||Ωi , then we have 1, , n Denote that ri ∇u p x |u|p x p− Ω ∇u n i − b0 |u|μ x p x |u|p x p− Ωi ∇u p x |u|p x p− ri >1 Ωi ∇u p x dx − b0 |u|μ x dx − b0 |u|μ x dx |u|p x − b0 |u|μ x p− ri ≤1 Ωi p |||u|||Ωi i ≤ ri >1 p− 3.24 dx n , p− μ − b0 kmi |||u|||Ωi− i where kmi infu∈Vm |Ωi , |||u|||Ωi Ωi |u|μ x dx As Vm |Ωi is a finite dimensional space, we have kmi > 0, for i 1, , n We denote by si the maximum of polynomial tpi /p− − b0 kmi tμi− on 0, ∞ , for i 1, , n Then there exists t0 > such that tpi − b0 kmi tμi− p− for t > t0 and i Let Rm |||v||| ≥ Rm /2 c8 ≤ 0, 3.25 n 1, , n, where c8 n/p− b1 meas Ω i si /p− max{2, p c8 1/ p− , 2nt0 } If ||z|| ≥ Rm , we get |||u||| ≥ Rm /2 or i If |||u||| ≥ Rm /2, |||u||| ≥ nt0 > It is easy to verify that there exists at least i0 such that |||u|||Ωi ≥ t0 > 1, thus pi I z ≤ |||u|||Ω0 i p− μi − b0 kmi0 |||u|||Ωi0 − c8 ≤ 3.26 10 Journal of Inequalities and Applications ii If |||v||| ≥ Rm /2, |||v||| ≥ p c8 1/p− 1/p− We obtain |||v|||p− − ≤ p− p I z ≤ c8 3.27 Now we get the result Lemma 3.5 There exist rm > and am → ∞ m → ∞ such that I z ≥ am , for any z ∈ X m−1 with ||z|| rm Proof For z ⊥ u, v ∈ X m−1 , v ⊥ By condition H2 , there exists c9 > such that F x, u, ≤ c9 |u|q1 x c9 3.28 Let ||z|| ≥ 1, we get ∇u I z ≥ |u|p x p x Ω ≥ p x ∇u p x |u|p x p Ω ∇u p x |u|p x p Ω − |u|α x − F x, u, α x − |u|α x − c9 |u|q1 α− − c10 |u|q1 x dx − c9 dx 3.29 dx − c11 x Denote that θm sup ⊥ u∈Vm Ω |u|q1 x dx, 3.30 |||u|||≤1 thus I z ≥ |||u|||p− − c10 θm u p q1 − c11 3.31 Let rm max 1, p− c10 p q1 θm 1/ q1 −p− , 2c11 p q1 q1 − p− 1/p− 3.32 Journal of Inequalities and Applications 11 By 5, Lemma 3.3 , we get that θm → 0, as m → ∞, then q1 − p− p q1 p I z ≥ rm− − c11 3.33 am , when m is sufficiently large and ||z|| rm It is easy to get that am → ∞, as m → ∞ Lemma 3.6 I is bounded from above on any bounded set of X m Proof For z u, v ∈ X m We get I z ≤ ∇u p x |u|p x p x Ω − F x, u, v dx 3.34 By conditions H2 and H3 , we know that if | u, v | ≥ R0 , F x, u, v ≥ and if | u, v | < R0 , |F x, u, v | ≤ c0 Then I z ≤ ∇u Ω p x p x |u|p x c12 dx, 3.35 and it is easy to get the result Proof of Theorem 1.1 By Lemmas 3.2–3.6 above, and 7, Proposition 2.1 and Remark 2.1 , we know that the functional I has a sequence of critical values ck → ∞, as k → ∞ Now we complete the proof Acknowledgments This work is supported by Science Research Foundation in Harbin Institute of Technology HITC200702 and The Natural Science Foundation of Heilongjiang Province A2007-04 References O Kov´ cik and J R´ kosn´k, “On spaces Lp x and W k,p x ,” Czechoslovak Mathematical Journal, vol aˇ a ı 41 116 , no 4, pp 592–618, 1991 M Ruˇ iˇ ka, Electrorheological Fluids: 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Ω p x p x un p x − ∇vn p x p x p x − H x, zn dx −→ ∞, 1.8 Journal of Inequalities and Applications Preliminaries First we recall some basic properties of variable exponent spaces Lp x Ω and variable... μ x ,p x μ x ε5 p x β x ≤ ε5 un β x μ x μ x − α x α x / α x −μ x ε3 μ x −α / μ−α ε3 μ x − , β x ? ?p x p x / ε5 β x ? ?p / β? ?p ε5 , μ x − p x p x / p x −μ x ε4 μ x ? ?p / μ? ?p ε4 β x − − p x −β x 3.6... and |u |p x inf λ > : Ω u λ p x dx ≤ 2.1 The variable exponent space Lp x Ω is the class of all functions u such that Ω |u x |p x dx < ∞ Under the assumption that p < ∞, Lp x Ω is a Banach space