Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 714179, 10 pages doi:10.1155/2009/714179 ResearchArticleInfinitelyManyPeriodicSolutionsforVariableExponent Systems Xiaoli Guo, 1 Mingxin Lu, 2 and Qihu Zhang 1 1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China 2 Department of Information Management, Nanjing University, Nanjing 210093, China Correspondence should be addressed to Qihu Zhang, zhangqh1999@yahoo.com.cn Received 4 December 2008; Accepted 14 April 2009 Recommended by Marta Garcia-Huidobro We mainly consider the system −Δ px u fvhu in R, −Δ qx v guωv in R,where 1 <px,qx ∈ C 1 R are periodic functions, and −Δ px u −|u | px−2 u is called px-Laplacian. We give the existence of infinitelymanyperiodicsolutions under some conditions. Copyright q 2009 Xiaoli Guo 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. 1. Introduction The study of differential equations and variational problems with variableexponent growth conditions has been a new and interesting topic. Many results have been obtained on this kind of problems, for example 1–18. On the applied background, we refer to 1, 3, 11, 18. In this paper, we mainly consider the existence of infinitelymanyperiodicsolutionsfor the system P ⎧ ⎨ ⎩ −Δ p x u f v h u in R, −Δ q x v g u ω v in R, 1.1 where px,qx ∈ C 1 R are functions. The operator −Δ px u −|u | px−2 u is called one- dimensional px-Laplacian. Especially, if px ≡ p a constant and qx ≡ q a constant, then P is the well-known constant exponent system. u, v is called a solution of P,ifu, v ∈ C 1 R, |u | px−2 u and |v | px−2 v are absolute continuous and satisfy P almost every where. 2 Journal of Inequalities and Applications In 19 , the authors consider the existence of positive weak solutionsfor the following constant exponent problems: I ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −Δ p u λf v , in Ω, −Δ p v λg u , in Ω, u v 0, on ∂Ω. 1.2 The first eigenfunction is used to construct the subsolution of constant exponent problems successfully. Under the condition that λ is large enough and lim u → ∞ f M g u 1/ p−1 u p−1 0, for every M>0, 1.3 the authors give the existence of positive solutionsfor problem I. In 20, the author considers the existence and nonexistence of positive weak solution to the following constant exponent elliptic system: II ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −Δ p u λu α v γ , in Ω, −Δ q v λu δ v β , in Ω, u v 0, on ∂Ω. 1.4 The first eigenfunction is used to construct the subsolution of constant exponent problems successfully. Because of the nonhomogeneity of px-Laplacian, px-Laplacian problems are more complicated than those of p-Laplacian. Maybe the first eigenvalue and the first eigenfunction of px-Laplacian do not exist see 6. Even if the first eigenfunction of px-Laplacian exists, because of the nonhomogeneity of px-Laplacian, the first eigenfunction cannot be used to construct the subsolution of px-Laplacian problems. There are many papers on the existence of periodicsolutionsfor p-Laplacian elliptic systems, for example 21–24. The results on the periodicsolutionsforvariableexponent systems are rare. Through a new method of constructing sub-supersolution, this paper gives the existence of infinitelymanyperiodicsolutionsfor problem P. 2. Main Results and Proofs At first, we give an existence of positive solutionsforvariableexponent systems on bounded domain via sub-super-solution method. The result itself has dependent value. Journal of Inequalities and Applications 3 Denote Ω R −R, R. Let us consider the existence of positive solutions of the following: P 1 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −Δ p x u f v h u , in Ω R , −Δ q x v g u ω v , in Ω R , u v 0, on ∂Ω R . 2.1 Write z sup x∈R zx, z − inf x∈R zx, for any z ∈ CR. Assume that H 1 px,qx ∈ C 1 R satisfy 1 <p − ≤ p < ∞, sup p x < ∞, 1 <q − ≤ q < ∞, sup q x < ∞. 2.2 H 2 f, g,h,ω : 0, ∞ → R are C 1 , monotone functions such that lim t → ∞ f t lim t → ∞ g t lim t → ∞ h t lim t → ∞ ω t ∞. 2.3 H 3 For any positive constant M, there are lim t → ∞ fMgt 1/q − −1 /t p − −1 0. H 4 lim t → ∞ ht/t p − −1 lim t → ∞ ωt/t q − −1 0. H 5 f, g, h,andω are odd functions such that f0g0h0ω00, px and qx are even, and T is a periodic of p and q, namely, pxpx T,qx qx T, for all x ∈ R. Note.In14, the present author discussed the existence of solutions of P 1 , under the conditions that P 1 is radial, pxqx,andh ω ≡ 0. Because of the nonhomogeneity of variableexponent problems, variableexponent problems are more complicated than constant exponent problems, and many results and methods for constant exponent problems are invalid forvariableexponent problems. In many cases, the radial symmetric conditions are effective to deal with variableexponent problems. There are many results about the radial variableexponent problems see 4, 14, 16, but the following Theorem 2.1 does not assume any symmetric conditions. We will establish. Theorem 2.1. If H 1 –H 4 hold, then P 1 possesses a positive solution, when R is sufficiently large. Proof. If we can construct a positive subsolution φ 1 ,φ 2 and supersolution z 1 ,z 2 of P 1 , namely, −Δ p x φ 1 ≤ f φ 2 h φ 1 , −Δ q x φ 2 ≤ g φ 1 ω φ 2 , for a.e. x ∈ R, −Δ p x z 1 ≥ f z 2 h z 1 , −Δ q x z 2 ≥ g z 1 ω z 2 , for a.e. x ∈ R, 2.4 which satisfy φ 1 ≤ z 1 and φ 2 ≤ z 2 , then P 1 possesses a positive solution see 5. 4 Journal of Inequalities and Applications Step 1. We will construct a subsolution of P 1 . Denfine φ 1 x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e −k 2 x−R − 1,R− a<x≤ R, e ak 2 − 1 R−a x k 2 e ak 2 p R−a −1 / p r −1 sin ε 2 r − R − a π 2 1/ p r −1 dr, R − a − π 2ε 2 <x≤ R − a, e ak 2 −1 R−a R−a−π/2ε 2 k 2 e ak 2 p R−a −1 / p r −1 sin ε 2 r− R−a π 2 1/ p r −1 dr, −R a π 2ε 1 <x≤ R − a − π 2ε 2 , e ak 1 − 1 x −Ra k 1 e ak 1 p −Ra −1 / p r −1 sin π 2 − ε 1 r − −R a 1/ p r −1 dr, −R a ≤ x ≤−R a π 2ε 1 , e k 1 xR − 1, −R ≤ x<−R a. φ 2 x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e −k 4 x−R − 1,R− a<x≤ R, e ak 4 − 1 R−a x k 4 e ak 4 q R−a −1 / q r −1 sin ε 4 r − R − a π 2 1/ q r −1 dr, R − a − π 2ε 4 <x≤ R − a, e ak 4 −1 R−a R−a−π/2ε 4 k 4 e ak 4 q R−a −1 / q r −1 sin ε 4 r− R−a π 2 1/ q r −1 dr, −R a π 2ε 3 <x≤ R − a − π 2ε 4 , e ak 3 − 1 x −Ra k 3 e ak 3 q −Ra −1 / q r −1 sin π 2 − ε 3 r − −R a 1/ q r −1 dr, −R a ≤ x ≤−R a π 2ε 3 , e k 3 xR − 1, −R ≤ x<−R a, 2.5 where a min inf p x − 1 4 sup p x 1 , inf q x − 1 4 sup q x 1 ,b min f 0 ,g 0 ,h 0 ,ω 0 , −1 , ε i k −p i e −ak i p ,i 1, 2,ε i k −q i e −ak i q ,i 3, 4; R> π ε i ,i 1, 2, 3, 4, 2.6 Journal of Inequalities and Applications 5 k 1 and k 2 satisfy e ak 2 − 1 R−a R−a−π/2ε 2 k 2 e ak 2 p R−a −1 / p r −1 sin ε 2 r − R − a π 2 1/ p r −1 dr e ak 1 −1 −Raπ/2ε 1 −Ra k 1 e ak 1 p −Ra −1 / p r −1 sin π 2 −ε 1 r− −Ra 1/ p r −1 dr, 2.7 k 3 and k 4 satisfy e ak 4 − 1 R−a R−a−π/2ε 4 k 4 e ak 4 q R−a −1/q r −1 sin ε 4 r − R − a π 2 1/q r −1 dr e ak 3 − 1 −Raπ/2ε 3 −Ra k 3 e ak 3 q −Ra −1/q r −1 sin π 2 − ε 3 r − −R a 1/q r −1 dr, 2.8 then φ 1 x ∈ C−R, R,andφ 2 x ∈ C−R, R. It is easy to see that φ i ≥ 0andφ i ∈ C 1 −R, R, i 1, 2. Obviously, ε i k −p i e −ak i p is continuous about k i . In the following, we will prove that φ 1 ,φ 2 is a subsolution for P 1 . By computation, − Δ p x φ 1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k 2 e −k 2 x−R p x −1 −k 2 p x −1 p x ln k 2 −k 2 p x x−R ,R− a<x≤ R, ε 2 k 2 e ak 2 p R−a −1 cos ε 2 x − R − a π 2 ,R− π 2ε 2 <x<R− a, 0, −Ra π 2ε 1 <x<R−a− π 2ε 2 , ε 1 k 1 e ak 1 p −Ra −1 cos π 2 − ε 1 x − −R a , −R a<x<−R a, k 1 e k 1 xR p x −1 −k 1 p x −1 −p x ln k 1 −k 1 p x xR , −R ≤ x<−R a. 2.9 If k 2 is sufficiently large, we have −Δ px φ 1 ≤−k 2 inf p x − 1 − sup p x ln k 2 k 2 R − r ≤−k 2 a, ∀x ∈ R − a, R . 2.10 As a is a constant and only depends on px and qx, when k 2 is large enough, we have −k 2 a<2b. Since φ 1 x ≥ 0andf h is monotone, we have −Δ p x φ 1 ≤ 2b ≤ f 0 h 0 ≤ f φ 2 h φ 1 ,R− a<x≤ R. 2.11 6 Journal of Inequalities and Applications According to H 2 , when k i are large enough, we have f e ak i − 1 ≥ 1,g e ak i − 1 ≥ 1,h e ak i − 1 ≥ 1,ω e ak i − 1 ≥ 1,i 1, 2, 3, 4, 2.12 where k i are dependent on f, g, h, ω, p and q, and they are independent on R. Since ε 2 k −p 2 e −ak 2 p , when x ∈ R − a − π/2ε, R − a, we have −Δ px φ 1 ε 2 k 2 e ak 2 p R−1 −1 cos ε x − R − a π 2 ≤ ε 2 k p 2 e ak 2 p 1. 2.13 Then we have −Δ p x φ 1 ≤ 1 ≤ f φ 2 h φ 1 ,R− a − π 2ε 2 <x<R− a. 2.14 Obviously −Δ p x φ 1 0 ≤ 1 ≤ f φ 2 h φ 1 , − R a π 2ε 1 <x<R− a − π 2ε 2 . 2.15 When k 2 is large enough, from 2.7 we can see that k 1 is large enough. Similar to the discussion of the above, we can conclude −Δ p x φ 1 ≤ 1 ≤ f φ 2 h φ 1 , − R a<x<−R a π 2ε 1 , 2.16 −Δ p x φ 1 ≤ f 0 h 0 ≤ f φ 2 h φ 1 , − R<x<−R a. 2.17 Since φ i x ∈ C 1 −R, R, combining 2.11, 2.14, 2.15, 2.16 and 2.17, we have −Δ p x φ 1 ≤ f φ 2 h φ 1 , for a.e. x ∈ −R, R . 2.18 Similarly, when k 4 is large enough, we have −Δ q x φ 2 ≤ g φ 1 ω φ 2 , for a.e. x ∈ −R, R . 2.19 Then φ 1 ,φ 2 is a subsolution of P 1 . Step 2. We will construct a supersolution of P 1 . Let z 1 be a solution of − z 1 p x −2 z 1 2μ, z 1 R 0 z 1 −R , 2.20 where μ is a positive constant and μ>1. Journal of Inequalities and Applications 7 Obviously, there exists x 0 ∈ Ω R such that z 1 x R x |r − x 0 2μ| 1/pr−1−1 2μr − x 0 dr. Note that x 0 is dependent on μ. Denote β β2μmax |x|≤R z 1 x. It is easy to see that 1 C μ 1/p −1 ≤ β 2μ ≤ Cμ 1/p − −1 , where C ≥ 1 is a positive constant. 2.21 Let us consider − p x z 1 2μ in Ω R , − q x z 2 2g β 2μ in Ω R , z 1 z 2 0on∂Ω R . 2.22 Similarly, we have 1 C g β 2μ 1/ q −1 ≤ max | x | ≤R z 2 x ≤ C g β 2μ 1/ q − −1 . 2.23 We will prove that z 1 ,z 2 is a supersolution for P 1 . From lim t → ∞ ωt/t q − −1 0and 2.23, when μ is large enough, we can easily see that −Δ q x z 2 2g β 2μ ≥ g z 1 ω z 2 . 2.24 Since lim t → ∞ fCg2t 1/q − −1 /t p − −1 0 and lim t → ∞ ht/t p − −1 0, when μ is large enough, according to 2.21 and 2.23, we have 2μ ≥ 2 1 C β 2μ p − −1 ≥ f C g β 2μ 1/ q − −1 h β 2μ . 2.25 This means that −Δ p x z 1 2μ ≥ f C g β 2μ 1/ q − −1 h β 2μ ≥ f z 2 h z 1 . 2.26 According to 2.24 and 2.26, we can conclude that z 1 ,z 2 is a supersolution for P 1 , when μ is large enough. Step 3. We will prove that φ 1 ≤ z 1 and φ 2 ≤ z 2 . Obviously, when μ is large enough, we can easily see that gβ2μ is large enough, then f φ 2 h φ 1 ≤ μ, ∀x ∈ Ω R , g φ 1 ω φ 2 ≤ g β 2μ , ∀x ∈ Ω R . 2.27 8 Journal of Inequalities and Applications Let us consider −Δ p x μ in Ω R . 2.28 It is easy to see that φ 1 is a subsolution of 2.28, when μ is large enough. Obviously, we can see that z 1 is a supersolution of 2.28,and z 1 R φ 1 R z 1 −R φ 1 −R 0. 2.29 According to the comparison principle see 12, we can see that φ 1 ≤ z 1 . Let us consider −Δ p x g β 2μ in Ω R . 2.30 It is easy to see that φ 2 is a subsolution of 2.30, when μ is large enough. Obviously, we can see that z 2 is a supersolution of 2.30,and z 2 R φ 2 R z 2 −R φ 2 −R 0. 2.31 According to the comparison principle see 12, we can see that φ 2 ≤ z 2 . Thus, we can conclude that φ 1 ≤ z 1 and φ 2 ≤ z 2 , when μ is sufficiently large. This completes the proof. Theorem 2.2. If H 1 –H 5 hold, then P has infinitelymanyperiodic solutions. Proof. Let R nT. According to Theorem 2.1, we can conclude that there exists an integer n 0 which is large enough such that P 1 has a positive solution u # n x,v # n x for any integer n ≥ n 0 . Since p and q are even, and f, g, h, and ω are odd, then −u # n −x, −v # n −x is a negative solution of P 1 . We can define a C 1 function u n x,v n x on −nT, 3nT as u n x ⎧ ⎨ ⎩ u # n x ,x∈ −nT, nT , −u # n − x − 2nT ,x∈ nT, 3nT , v n x ⎧ ⎨ ⎩ v # n x ,x∈ −nT, nT , −v # n − x − 2nT ,x∈ nT, 3nT . 2.32 We extend u n x,v n x as u n x,v n xu n x m4nT,v n x m4nT, where m is an integer such that x m4nT ∈ −nT, 3nT. It is easy to see that u n ,v n ∈ C 1 R, u n x,v n x is a solution of P, and the periodic of u n x,v n x is 4nT. This completes the proof. 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Ge, “Periodic solutions of non-autonomous second-order systems with a p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 1, pp. 192–203, 2007. . Inequalities and Applications Volume 2009, Article ID 714179, 10 pages doi:10.1155/2009/714179 Research Article Infinitely Many Periodic Solutions for Variable Exponent Systems Xiaoli Guo, 1 Mingxin. problems. There are many papers on the existence of periodic solutions for p-Laplacian elliptic systems, for example 21–24. The results on the periodic solutions for variable exponent systems. of variable exponent problems, variable exponent problems are more complicated than constant exponent problems, and many results and methods for constant exponent problems are invalid for variable