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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 434938, 18 pages
http://dx.doi.org/10.1155/2012/434938
Research Article

Existence of Subharmonic Solutions for a Class of Second-Order 𝑝-Laplacian Systems with Impulsive Effects

1Department of Mathematics and Computation Science, Yulin Normal University, Yulin, Guangxi 537000, China
2College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China
3School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received 14 April 2011; Accepted 1 July 2011

Academic Editor: MengΒ Fan

Copyright Β© 2012 Wen-Zhen Gong 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.

Abstract

By using minimax methods in critical point theory, a new existence theorem of infinitely many periodic solutions is obtained for a class of second-order 𝑝-Laplacian systems with impulsive effects. Our result generalizes many known works in the literature.

1. Introduction

Consider the following 𝑝-Laplacian system with impulsive effects: 𝑑||||𝑑𝑑̇𝑒(𝑑)π‘βˆ’2||||̇𝑒(𝑑)βˆ’πΏ(𝑑)𝑒(𝑑)π‘βˆ’2𝑒(𝑑)+βˆ‡πΉ(𝑑,𝑒(𝑑))=0,a.e.Ξ”ξ‚€||ξ€·π‘‘π‘‘βˆˆβ„,𝑒(0)βˆ’π‘’(𝑇)=̇𝑒(0)βˆ’Μ‡π‘’(𝑇)=0,̇𝑒𝑗||π‘βˆ’2𝑑̇𝑒𝑗=|||𝑑̇𝑒+𝑗|||π‘βˆ’2𝑑̇𝑒+π‘—ξ‚βˆ’||ξ€·π‘‘Μ‡π‘’βˆ’π‘—ξ€Έ||π‘βˆ’2ξ€·π‘‘Μ‡π‘’βˆ’π‘—ξ€Έ=βˆ‡πΌπ‘—ξ€·π‘’ξ€·π‘‘π‘—ξ€Έξ€Έ,𝑗=1,2,…,π‘š,(1.1) where 𝑝>1, 𝑇>0, 0=𝑑0<𝑑1<𝑑2<β‹―<π‘‘π‘š<π‘‘π‘š+1=𝑇, and βˆ‡πΌπ‘—βˆΆβ„π‘β†’β„π‘(𝑗=1,2,…,π‘š) are continuous and πΉβˆΆβ„Γ—β„π‘β†’β„ is 𝑇-periodic in 𝑑 for all π‘’βˆˆβ„π‘, βˆ‡πΉ(𝑑,𝑒) is the gradient of 𝐹(𝑑,𝑒) with respect to 𝑒. 𝐿∈𝐢(ℝ,ℝ𝑁×𝑁) is a 𝑇-periodic positive definite symmetric matrix.

Throughout this paper, we always assume the following condition holds.(A)𝐹(𝑑,π‘₯) is measurable in 𝑑 for all π‘₯βˆˆβ„π‘ and continuously differentiable in π‘₯ for a.e. π‘‘βˆˆ[0,𝑇], and there exist π‘ŽβˆˆπΆ(ℝ+,ℝ+), π‘βˆˆπΏ1([0,𝑇];ℝ+) such that ||||||||𝐹(𝑑,π‘₯)β‰€π‘Ž(|π‘₯|)𝑏(𝑑),βˆ‡πΉ(𝑑,π‘₯)β‰€π‘Ž(|π‘₯|)𝑏(𝑑)(1.2) for all π‘₯βˆˆβ„π‘ and a.e. π‘‘βˆˆ[0,𝑇].

For the sake of convenience, in the sequel, we define 𝐡={1,2,…,π‘š}.

When 𝑝=2, βˆ‡πΌπ‘—β‰‘0, π‘—βˆˆπ΅, problem (1.1) becomes the following second-order Hamiltonian system: Μˆπ‘’(𝑑)βˆ’πΏ(𝑑)𝑒(𝑑)+βˆ‡πΉ(𝑑,𝑒(𝑑))=0,a.e.π‘‘βˆˆβ„.(1.3)

There are many papers concerning the existence of periodic solutions or homoclinic solutions for problem (1.3) by minimax methods. Here for identifying a few, we only mention [1–8].

For βˆ‡πΌπ‘—β‰ 0, π‘—βˆˆπ΅, problem (1.1) involves impulsive effects. Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is jumps) in their values. Since these processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the processes, it is natural to suppose that these perturbations act instantaneously, that is, in the form of impulse. Processes of this type are often investigated in various fields of science and technology, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, and so on. For more details of impulsive differential equations, we refer the readers to the books [9, 10].

There are many methods for finding periodic solutions of impulsive differential equations, such as the monotone-iterative technique, a numerical-analytical method, the method of upper and lower solutions, and the method of bilateral approximations. For more information about periodic solutions of impulsive differential equations, one can refer to the papers [11–18]. However, there are few papers [19–25] concerning periodic solutions of impulsive differential equations by variational methods. So it is a novel method to employ variational methods to investigate the existence of periodic solutions for impulsive differential equations.

Motivated by the above papers, we study the existence of subharmonic solutions for problem (1.1) by applying minimax methods in critical point theory. Our result is new, which seems not to be found in the literature.

Throughout this paper, let π‘žβˆˆ(1,+∞) satisfy 1/𝑝+1/π‘ž=1.

2. Preliminaries

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we construct a variational structure. With this variational structure, we can reduce the problem of finding solutions of problem (1.1) to that of seeking the critical points of the corresponding functional.

Let π‘˜ be a positive integer and π‘Š1,π‘π‘˜π‘‡ the Sobolev space defined byπ‘Š1,π‘π‘˜π‘‡=ξ€½π‘’βˆΆβ„β†’β„π‘βˆ£π‘’isabsolutelycontinuous,𝑒(𝑑)=𝑒(𝑑+π‘˜π‘‡),Μ‡π‘’βˆˆπΏπ‘ξ€·[]0,π‘˜π‘‡;ℝ𝑁(2.1) with the norm β€–ξ‚΅ξ€œπ‘’β€–=0π‘˜π‘‡||||𝑒(𝑑)π‘ξ€œπ‘‘π‘‘+0π‘˜π‘‡||||̇𝑒(𝑑)𝑝𝑑𝑑1/𝑝.(2.2)

Take π‘£βˆˆπ‘Š1,π‘π‘˜π‘‡ and multiply the two sides of the equality𝑑||||𝑑𝑑̇𝑒(𝑑)π‘βˆ’2||||̇𝑒(𝑑)βˆ’πΏ(𝑑)𝑒(𝑑)π‘βˆ’2𝑒(𝑑)+βˆ‡πΉ(𝑑,𝑒(𝑑))=0(2.3) by 𝑣 and integrate from 0 to π‘˜π‘‡; we haveξ€œ0π‘˜π‘‡ξ‚΅ξ‚€||||̇𝑒(𝑑)π‘βˆ’2̇𝑒(𝑑)ξ…žξ‚Άξ€œ,𝑣(𝑑)𝑑𝑑=0π‘˜π‘‡ξ‚€||||𝐿(𝑑)𝑒(𝑑)π‘βˆ’2ξ‚ξ€œπ‘’(𝑑),𝑣(𝑑)π‘‘π‘‘βˆ’0π‘˜π‘‡(βˆ‡πΉ(𝑑,𝑒(𝑑)),𝑣(𝑑))𝑑𝑑.(2.4) Moreover, by ̇𝑒(0)=̇𝑒(𝑇), one hasξ€œ0π‘˜π‘‡ξ‚΅ξ‚€||||̇𝑒(𝑑)π‘βˆ’2̇𝑒(𝑑)ξ…žξ‚Άξ€œ,𝑣(𝑑)𝑑𝑑=π‘˜π‘‡0ξ‚΅ξ‚€||||̇𝑒(𝑑)π‘βˆ’2̇𝑒(𝑑)ξ…žξ‚Ά,𝑣(𝑑)𝑑𝑑=π‘˜π‘šξ“π‘—=0ξ€œπ‘‘π‘—+1𝑑𝑗||||̇𝑒(𝑑)π‘βˆ’2̇𝑒(𝑑)ξ…žξ‚Ά,𝑣(𝑑)𝑑𝑑=π‘˜π‘šξ“π‘—=0|||ξ‚€π‘‘Μ‡π‘’βˆ’π‘—+1|||π‘βˆ’2ξ‚€π‘‘Μ‡π‘’βˆ’π‘—+1ξ‚π‘£ξ‚€π‘‘βˆ’π‘—+1ξ‚βˆ’|||𝑑̇𝑒+𝑗|||π‘βˆ’2𝑑̇𝑒+𝑗𝑣𝑑+π‘—ξ‚βˆ’ξ€œπ‘‘π‘—+1𝑑𝑗||||̇𝑒(𝑑)π‘βˆ’2̇̇𝑒(𝑑),𝑣(𝑑)𝑑𝑑=π‘˜π‘šξ“π‘—=0ξ‚΅|||ξ‚€π‘‘Μ‡π‘’βˆ’π‘—+1|||π‘βˆ’2ξ‚€π‘‘Μ‡π‘’βˆ’π‘—+1ξ‚π‘£ξ‚€π‘‘βˆ’π‘—+1ξ‚βˆ’|||𝑑̇𝑒+𝑗|||π‘βˆ’2𝑑̇𝑒+𝑗𝑣𝑑+π‘—ξ‚ξ‚Άβˆ’ξ€œ0π‘˜π‘‡ξ‚€||||̇𝑒(𝑑)π‘βˆ’2̇||||̇𝑒(𝑑),𝑣(𝑑)𝑑𝑑=π‘˜Μ‡π‘’(𝑇)π‘βˆ’2||||̇𝑒(𝑇)𝑣(𝑇)βˆ’π‘˜Μ‡π‘’(0)π‘βˆ’2̇𝑒(0)𝑣(0)βˆ’π‘˜π‘šξ“π‘—=1βˆ‡πΌπ‘—ξ€·π‘’ξ€·π‘‘π‘—π‘£ξ€·π‘‘ξ€Έξ€Έπ‘—ξ€Έβˆ’ξ€œ0π‘˜π‘‡ξ‚€||||̇𝑒(𝑑)π‘βˆ’2̇̇𝑒(𝑑),𝑣(𝑑)𝑑𝑑=βˆ’π‘˜π‘šξ“π‘—=1βˆ‡πΌπ‘—ξ€·π‘’ξ€·π‘‘π‘—π‘£ξ€·π‘‘ξ€Έξ€Έπ‘—ξ€Έβˆ’ξ€œ0π‘˜π‘‡ξ‚€||||̇𝑒(𝑑)π‘βˆ’2̇̇𝑒(𝑑),𝑣(𝑑)𝑑𝑑.(2.5) Together with (2.4), we get ξ€œ0π‘˜π‘‡ξ‚€||||̇𝑒(𝑑)π‘βˆ’2̇̇𝑒(𝑑),𝑣(𝑑)𝑑𝑑+π‘˜π‘šξ“π‘—=1βˆ‡πΌπ‘—ξ€·π‘’ξ€·π‘‘π‘—π‘£ξ€·π‘‘ξ€Έξ€Έπ‘—ξ€Έ+ξ€œ0π‘˜π‘‡ξ‚€||||𝐿(𝑑)𝑒(𝑑)π‘βˆ’2=ξ€œπ‘’(𝑑),𝑣(𝑑)𝑑𝑑0π‘˜π‘‡(βˆ‡πΉ(𝑑,𝑒(𝑑)),𝑣(𝑑))𝑑𝑑.(2.6)

Definition 2.1. We say that a function π‘’βˆˆπ‘Š1,π‘π‘˜π‘‡ is a weak solution of problem (1.1) if the identity ξ€œ0π‘˜π‘‡ξ‚€||||̇𝑒(𝑑)π‘βˆ’2̇̇𝑒(𝑑),𝑣(𝑑)𝑑𝑑+π‘˜π‘šξ“π‘—=1βˆ‡πΌπ‘—ξ€·π‘’ξ€·π‘‘π‘—π‘£ξ€·π‘‘ξ€Έξ€Έπ‘—ξ€Έ+ξ€œ0π‘˜π‘‡ξ‚€||||𝐿(𝑑)𝑒(𝑑)π‘βˆ’2=ξ€œπ‘’(𝑑),𝑣(𝑑)𝑑𝑑0π‘˜π‘‡(βˆ‡πΉ(𝑑,𝑒(𝑑)),𝑣(𝑑))𝑑𝑑(2.7) holds for any π‘£βˆˆπ‘Š1,π‘π‘˜π‘‡.

Define the functional πœ™π‘˜ on π‘Š1,π‘π‘˜π‘‡ by πœ™π‘˜1(𝑒)=π‘ξ€œ0π‘˜π‘‡ξ‚ƒ||||̇𝑒(𝑑)𝑝+ξ‚€||||𝐿(𝑑)𝑒(𝑑)π‘βˆ’2ξ€œπ‘’(𝑑),𝑒(𝑑)ξ‚ξ‚„π‘‘π‘‘βˆ’0π‘˜π‘‡πΉ(𝑑,𝑒(𝑑))𝑑𝑑+π‘˜π‘šξ“π‘—=1𝐼𝑗𝑒𝑑𝑗=πœ‘π‘˜(𝑒)+πœ“π‘˜(𝑒),π‘’βˆˆπ‘Š1,π‘π‘˜π‘‡,(2.8) where πœ‘π‘˜1(𝑒)=π‘ξ€œ0π‘˜π‘‡ξ‚ƒ||||̇𝑒(𝑑)𝑝+ξ‚€||||𝐿(𝑑)𝑒(𝑑)π‘βˆ’2ξ€œπ‘’(𝑑),𝑒(𝑑)ξ‚ξ‚„π‘‘π‘‘βˆ’0π‘˜π‘‡πœ“πΉ(𝑑,𝑒(𝑑))𝑑𝑑,π‘˜(𝑒)=π‘˜π‘šξ“π‘—=1𝐼𝑗𝑒𝑑𝑗.ξ€Έξ€Έ(2.9)

It follows from assumption (A) that the functional πœ‘π‘˜ is continuously differentiable on π‘Š1,π‘π‘˜π‘‡ andβŸ¨πœ‘π‘˜ξ€œβ€²(𝑒),π‘£βŸ©=0π‘˜π‘‡||||̇𝑒(𝑑)π‘βˆ’2̇+ξ‚€||||̇𝑒(𝑑),𝑣(𝑑)𝐿(𝑑)𝑒(𝑑)π‘βˆ’2𝑒(𝑑),𝑣(𝑑)βˆ’(βˆ‡πΉ(𝑑,𝑒(𝑑)),𝑣(𝑑))𝑑𝑑(2.10) for 𝑒,π‘£βˆˆπ‘Š1,π‘π‘˜π‘‡. By the continuity of βˆ‡πΌπ‘—, π‘—βˆˆπ΅, one has that πœ“π‘˜βˆˆ(π‘Š1,π‘π‘˜π‘‡,ℝ). Hence, πœ™π‘˜(𝑒)∈(π‘Š1,π‘π‘˜π‘‡,ℝ). For any π‘£βˆˆπ‘Š1,π‘π‘˜π‘‡, we have βŸ¨πœ™π‘˜ξ€œβ€²(𝑒),π‘£βŸ©=0π‘˜π‘‡||||̇𝑒(𝑑)π‘βˆ’2̇+ξ‚€||||̇𝑒(𝑑),𝑣(𝑑)𝐿(𝑑)𝑒(𝑑)π‘βˆ’2𝑒(𝑑),𝑣(𝑑)βˆ’(βˆ‡πΉ(𝑑,𝑒(𝑑)),𝑣(𝑑))𝑑𝑑+π‘˜π‘šξ“π‘—=1βˆ‡πΌπ‘—ξ€·π‘’ξ€·π‘‘π‘—π‘£ξ€·π‘‘ξ€Έξ€Έπ‘—ξ€Έ.(2.11)

By Definition 2.1, the weak solutions of problem (1.1) correspond to the critical points of the functional πœ™π‘˜.

For π‘’βˆˆπ‘Š1,π‘π‘˜π‘‡, let βˆ«π‘’=(1/π‘˜π‘‡)0π‘˜π‘‡π‘’(𝑑)𝑑𝑑 and ̃𝑒(𝑑)=𝑒(𝑑)βˆ’π‘’; then it follows from Proposition  1.1 in [26] that β€–π‘’β€–βˆžβˆΆ=max[]π‘‘βˆˆ0,π‘˜π‘‡||||≀𝑒(𝑑)(π‘˜π‘‡)βˆ’1/𝑝+(π‘˜π‘‡)1/π‘žξ€Έβ€–π‘’β€–=π‘‘π‘˜β€–π‘’β€–,(2.12) where π‘‘π‘˜=(π‘˜π‘‡)βˆ’1/𝑝+(π‘˜π‘‡)1/π‘ž, and if ∫(1/π‘˜π‘‡)0π‘˜π‘‡π‘’(𝑑)𝑑𝑑=0, then β€–Μƒπ‘’β€–βˆžβˆΆ=max[]π‘‘βˆˆ0,π‘˜π‘‡||||̃𝑒(𝑑)≀(π‘˜π‘‡)1/π‘žβ€–Μ‡π‘’β€–πΏπ‘,(2.13)‖̃𝑒‖𝑝𝐿𝑝≀(π‘˜π‘‡)𝑝‖̇𝑒‖𝑝𝐿𝑝,(2.14) where 1/𝑝+1/π‘ž=1. Let ξ‚‹π‘Š1,π‘π‘˜π‘‡={π‘’βˆˆπ‘Š1,π‘π‘˜π‘‡βˆ£π‘’=0}; then π‘Š1,π‘π‘˜π‘‡=ξ‚‹π‘Š1,π‘π‘˜π‘‡βŠ•β„π‘. We will use the following lemma to prove our main results.

Lemma 2.2 (see [27]). Let 𝐸 be a real Banach space with 𝐸=𝑋1βŠ•π‘‹2, where 𝑋1 is finite dimensional. Suppose that πœ‘βˆˆπΆ1(𝐸,ℝ) satisfies the (PS) condition, and (a)there exist constants 𝜌, 𝛼>0 such that πœ‘|πœ•π΅πœŒβ‹‚π‘‹2β‰₯𝛼, where 𝐡𝜌∢={π‘’βˆˆπΈβˆ£β€–π‘’β€–β‰€πœŒ}, and πœ•π΅πœŒ denotes the boundary of 𝐡𝜌; (b)there exists an π‘’βˆˆπœ•π΅1⋂𝑋2 and 𝐿>𝜌 such that if 𝑄≑(𝐡𝐿⋂𝑋1)βŠ•{π‘Ÿπ‘’βˆ£0β‰€π‘Ÿβ‰€πΏ}, then πœ‘|πœ•π‘„β‰€0.
Then πœ‘ possesses a critical value 𝑐β‰₯𝛼 which can be characterized as 𝑐=infβ„ŽβˆˆΞ“maxπ‘’βˆˆπ‘„πœ‘(β„Ž(𝑒)), where Ξ“={β„ŽβˆˆπΆ(𝑄,𝐸)βˆ£β„Ž=idonπœ•π‘„}.

It is well known that a deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition. So Lemma 2.2 holds true under condition (C).

3. Main Result and Proof

Theorem 3.1. Assume that (A) holds and 𝐹,𝐼𝑗 satisfy the following conditions: (I1)there exists 𝑐𝑗>0 such that 0≀𝐼𝑗(𝑐π‘₯)β‰€π‘—π‘˜|π‘₯|𝑝,π‘—βˆˆπ΅,βˆ€π‘₯βˆˆβ„π‘;(3.1)(I2)for any π‘—βˆˆπ΅, βˆ‡πΌπ‘—(π‘₯)π‘₯≀𝑝𝐼𝑗(π‘₯),βˆ€π‘₯βˆˆβ„π‘;(3.2)(H1)βˆ«π‘‡0𝐹(𝑑,π‘₯)𝑑𝑑β‰₯0, for all π‘₯βˆˆβ„π‘;(H2)lim|π‘₯|β†’0(𝐹(𝑑,π‘₯)/|π‘₯|𝑝)=0 uniformly for a.e. π‘‘βˆˆ[0,𝑇];(H3)lim|π‘₯|β†’βˆž(𝐹(𝑑,π‘₯)/|π‘₯|𝑝)=+∞ uniformly for a.e. π‘‘βˆˆ[0,𝑇];(H4)there exists a positive constant 𝑀 such that limsup|π‘₯|β†’βˆž(𝐹(𝑑,π‘₯)/|π‘₯|π‘Ÿ)≀𝑀 uniformly for a.e. π‘‘βˆˆ[0,𝑇];(H5)there exists 𝑀1>0 such that liminf|π‘₯|β†’βˆž((βˆ‡πΉ(𝑑,π‘₯),π‘₯)βˆ’π‘πΉ(𝑑,π‘₯))/|π‘₯|πœ‡β‰₯𝑀1 uniformly for a.e. π‘‘βˆˆ[0,𝑇],
where π‘Ÿ>𝑝 and πœ‡>π‘Ÿβˆ’π‘. Then problem (1.1) has a sequence of distinct periodic solutions with period π‘˜π‘—π‘‡ satisfying π‘˜π‘—βˆˆβ„• and π‘˜π‘—β†’βˆž as π‘—β†’βˆž.

Remark 3.2. As far as we know, there is no paper considering subharmonic solutions of impulsive differential equations. Our result is new.

Proof. The proof is divided into three steps. In the following, 𝐢𝑖(𝑖=1,…) denote different positive constants.Step 1. The functional πœ™π‘˜ satisfies condition (C). Let {𝑒𝑛}βŠ‚π‘Š1,π‘π‘˜π‘‡ satisfying (1+‖𝑒𝑛‖)β€–πœ™ξ…žπ‘˜(𝑒𝑛)β€–β†’0 as π‘›β†’βˆž and πœ™π‘˜(𝑒𝑛) is bounded; then, there exists a constant 𝐢1 such that ||πœ™π‘˜ξ€·π‘’π‘›ξ€Έ||≀𝐢1,‖‖𝑒1+π‘›β€–β€–ξ€Έβ€–β€–πœ™ξ…žπ‘˜ξ€·π‘’π‘›ξ€Έβ€–β€–β‰€πΆ1.(3.3) From (H4), there exists 𝑀2>0 such that 𝐹(𝑑,π‘₯)≀𝑀|π‘₯|π‘Ÿβˆ€|π‘₯|β‰₯𝑀2,a.e.[].π‘‘βˆˆ0,𝑇(3.4) By assumption (A), for |π‘₯|≀𝑀2, there exists 𝐢2=max|π‘₯|≀𝑀2π‘Ž(|π‘₯|)>0 such that ||||𝐹(𝑑,π‘₯)≀𝐢2𝑏(𝑑),(3.5) which together with (3.4) implies that 𝐹(𝑑,π‘₯)≀𝑀|π‘₯|π‘Ÿ+𝐢2𝑏(𝑑),βˆ€π‘₯βˆˆβ„π‘,a.e.[].π‘‘βˆˆ0,𝑇(3.6) By (3.3) and (3.6), we have πœ™π‘˜ξ€·π‘’π‘›ξ€Έ+ξ€œ0π‘˜π‘‡πΉξ€·π‘‘,𝑒𝑛𝑑𝑑≀𝐢1+ξ€œ0π‘˜π‘‡ξ€·π‘€||𝑒𝑛||(𝑑)π‘Ÿ+𝐢2𝑏(𝑑)𝑑𝑑=𝐢1+𝐢2π‘˜β€–π‘β€–πΏ1ξ€œ+𝑀0π‘˜π‘‡||𝑒𝑛||(𝑑)π‘Ÿπ‘‘π‘‘=𝐢3ξ€œ+𝑀0π‘˜π‘‡||𝑒𝑛||(𝑑)π‘Ÿπ‘‘π‘‘.(3.7) Since 𝐿(𝑑) is continuous 𝑇-periodic positive definite symmetric matrix on [0,𝑇], there exist constants 𝑐1,𝑐2>0 such that 𝑐1|π‘₯|𝑝≀𝐿(𝑑)|π‘₯|π‘βˆ’2ξ€Έπ‘₯,π‘₯≀𝑐2|π‘₯|𝑝,βˆ€π‘₯βˆˆβ„π‘.(3.8) It follows from (3.8) and (I1) that πœ™π‘˜ξ€·π‘’π‘›ξ€Έ+ξ€œ0π‘˜π‘‡πΉξ€·π‘‘,𝑒𝑛1𝑑𝑑=π‘ξ€œ0π‘˜π‘‡ξ‚ƒ||̇𝑒𝑛||(𝑑)𝑝+ξ‚€||𝑒𝐿(𝑑)𝑛||(𝑑)π‘βˆ’2𝑒𝑛(𝑑),𝑒𝑛(𝑑)𝑑𝑑+π‘˜π‘šξ“π‘—=1𝐼𝑗𝑒𝑑𝑗β‰₯1ξ€Έξ€Έπ‘ξ€œ0π‘˜π‘‡ξ€Ί||̇𝑒𝑛(||𝑑)𝑝+𝑐1||𝑒𝑛(||𝑑)𝑝1𝑑𝑑β‰₯min𝑝,𝑐1𝑝‖‖𝑒𝑛‖‖𝑝=𝐢4‖‖𝑒𝑛‖‖𝑝.(3.9) By (3.7) and (3.9), we get 𝐢4‖‖𝑒𝑛‖‖𝑝≀𝐢3ξ€œ+𝑀0π‘˜π‘‡||𝑒𝑛||(𝑑)π‘Ÿπ‘‘π‘‘.(3.10) From (H5), there exists 𝑀3>0 such that (βˆ‡πΉ(𝑑,π‘₯),π‘₯)βˆ’π‘πΉ(𝑑,π‘₯)β‰₯𝑀1|π‘₯|πœ‡for|π‘₯|β‰₯𝑀3,a.e.[].π‘‘βˆˆ0,𝑇(3.11) By assumption (A), for |π‘₯|≀𝑀3, there exists 𝐢5=max|π‘₯|≀𝑀3π‘Ž(|π‘₯|)>0 such that ||||(βˆ‡πΉ(𝑑,π‘₯),π‘₯)βˆ’π‘πΉ(𝑑,π‘₯)≀𝐢5𝑝+𝑀3𝑏(𝑑).(3.12) Thus, from (3.11) and (3.12), we have (βˆ‡πΉ(𝑑,π‘₯),π‘₯)βˆ’π‘πΉ(𝑑,π‘₯)β‰₯𝑀1|π‘₯|πœ‡βˆ’π‘€1π‘€πœ‡3βˆ’πΆ5𝑝+𝑀3𝑏(𝑑)forπ‘₯βˆˆβ„π‘,a.e.[],π‘‘βˆˆ0,𝑇(3.13) which together with (3.3) and (I2) implies that (𝑝+1)𝐢1β‰₯π‘πœ™π‘˜ξ€·π‘’π‘›ξ€Έβˆ’ξ«πœ™π‘˜β€²ξ€·π‘’π‘›ξ€Έ,𝑒𝑛=ξ€œ0π‘˜π‘‡ξ€·ξ€Ίξ€·βˆ‡πΉπ‘‘,𝑒𝑛,π‘’π‘›ξ€Έξ€·βˆ’π‘πΉπ‘‘,𝑒𝑛𝑑𝑑+π‘π‘˜π‘šξ“π‘—=1πΌπ‘—ξ€·π‘’π‘›ξ€·π‘‘π‘—ξ€Έξ€Έβˆ’π‘˜π‘šξ“π‘—=1βˆ‡πΌπ‘—ξ€·π‘’π‘›ξ€·π‘‘π‘—π‘’ξ€Έξ€Έπ‘›ξ€·π‘‘π‘—ξ€Έβ‰₯𝑀1ξ€œ0π‘˜π‘‡||𝑒𝑛||(𝑑)πœ‡π‘‘π‘‘βˆ’πΆ5𝑝+𝑀3ξ€Έξ€œ0π‘˜π‘‡π‘(𝑑)π‘‘π‘‘βˆ’π‘€1π‘€πœ‡3π‘˜π‘‡=𝑀1ξ€œ0π‘˜π‘‡||𝑒𝑛||(𝑑)πœ‡π‘‘π‘‘βˆ’πΆ6.(3.14) Hence, ∫0π‘˜π‘‡|𝑒𝑛(𝑑)|πœ‡π‘‘π‘‘ is bounded. If πœ‡>π‘Ÿ, we have ξ€œ0π‘˜π‘‡||𝑒𝑛||(𝑑)π‘Ÿπ‘‘π‘‘β‰€(π‘˜π‘‡)(πœ‡βˆ’π‘Ÿ)/πœ‡ξ‚΅ξ€œ0π‘˜π‘‡||𝑒𝑛||(𝑑)πœ‡ξ‚Άπ‘‘π‘‘π‘Ÿ/πœ‡,(3.15) which together with (3.10) implies that ‖𝑒𝑛‖ is bounded. If πœ‡β‰€π‘Ÿ, then from (2.12), we get ξ€œ0π‘˜π‘‡||𝑒𝑛||(𝑑)π‘Ÿβ€–β€–π‘’π‘‘π‘‘β‰€π‘›β€–β€–βˆžπ‘Ÿβˆ’πœ‡ξ‚΅ξ€œ0π‘˜π‘‡||𝑒𝑛||(𝑑)πœ‡ξ‚Άπ‘‘π‘‘π‘Ÿ/πœ‡β‰€π‘‘π‘˜π‘Ÿβˆ’πœ‡β€–β€–π‘’π‘›β€–β€–π‘Ÿβˆ’πœ‡ξ‚΅ξ€œ0π‘˜π‘‡||𝑒𝑛||(𝑑)πœ‡ξ‚Άπ‘‘π‘‘π‘Ÿ/πœ‡.(3.16) Since πœ‡>π‘Ÿβˆ’π‘, it follows from (3.10) that ‖𝑒𝑛‖ is bounded too. Therefore, ‖𝑒𝑛‖ is bounded in π‘Š1,π‘π‘˜π‘‡. Hence, there exists a subsequence, still denoted by {𝑒𝑛}, such that 𝑒𝑛⇀𝑒0weaklyinπ‘Š1,π‘π‘˜π‘‡,(3.17)π‘’π‘›βŸΆπ‘’0stronglyin𝐢[]0,π‘˜π‘‡;ℝ𝑁,(3.18)π‘’π‘›βŸΆπ‘’0stronglyin𝐿𝑝[]0,π‘˜π‘‡;ℝ𝑁.(3.19) From (2.11), we have ξ«πœ™π‘˜β€²ξ€·π‘’π‘›ξ€Έ,π‘’π‘›βˆ’π‘’0=ξ€œ0π‘˜π‘‡||̇𝑒𝑛||(𝑑)π‘βˆ’2̇𝑒𝑛(𝑑),̇𝑒𝑛(𝑑)βˆ’Μ‡π‘’0+ξ‚€||𝑒(𝑑)𝐿(𝑑)𝑛||(𝑑)π‘βˆ’2𝑒𝑛(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0βˆ’ξ€œ(𝑑)𝑑𝑑0π‘˜π‘‡ξ€·ξ€·βˆ‡πΉπ‘‘,𝑒𝑛(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0ξ€Έ(𝑑)𝑑𝑑+π‘˜π‘šξ“π‘—=1ξ€·βˆ‡πΌπ‘—ξ€·π‘’π‘›ξ€·π‘‘π‘—ξ€Έξ€Έ,π‘’π‘›ξ€·π‘‘π‘—ξ€Έβˆ’π‘’0𝑑𝑗.ξ€Έξ€Έ(3.20) From (3.3) and (3.18), we have ||ξ«πœ™ξ…žπ‘˜ξ€·π‘’π‘›ξ€Έ,π‘’π‘›βˆ’π‘’0||β‰€β€–β€–πœ™ξ…žπ‘˜ξ€·π‘’π‘›ξ€Έβ€–β€–β€–β€–π‘’π‘›βˆ’π‘’0β€–β€–βŸΆ0asπ‘›βŸΆβˆž.(3.21) By (3.8), we know that 𝑐1≀‖𝐿‖≀𝑐2, which together with the boundedness of {𝑒𝑛} and (3.19) implies that ξ€œ0π‘˜π‘‡ξ‚€||𝑒𝐿(𝑑)𝑛||(𝑑)π‘βˆ’2𝑒𝑛(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0‖‖𝑒(𝑑)π‘‘π‘‘β‰€β€–πΏβ€–π‘›β€–β€–πΏπ‘βˆ’1π‘β€–β€–π‘’π‘›βˆ’π‘’0β€–β€–πΏπ‘βŸΆ0asπ‘›βŸΆβˆž.(3.22) From the boundedness of {𝑒𝑛}, the continuity of βˆ‡πΌπ‘—, and (3.18), we have π‘šξ“π‘—=1ξ€·βˆ‡πΌπ‘—ξ€·π‘’π‘›ξ€·π‘‘π‘—ξ€Έξ€Έ,π‘’π‘›ξ€·π‘‘π‘—ξ€Έβˆ’π‘’0ξ€·π‘‘π‘—ξ€Έξ€ΈβŸΆ0asπ‘›βŸΆβˆž.(3.23) It follows from (A), (3.18) and the boundedness of {𝑒𝑛} that ξ€œ0π‘˜π‘‡ξ€·ξ€·βˆ‡πΉπ‘‘,𝑒𝑛(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0ξ€Έ(𝑑)π‘‘π‘‘βŸΆ0asπ‘›βŸΆβˆž,(3.24) which together with (3.20), (3.21), (3.22), and (3.23) implies that ξ€œ0π‘˜π‘‡ξ‚€||̇𝑒𝑛||(𝑑)π‘βˆ’2̇𝑒𝑛(𝑑),̇𝑒𝑛(𝑑)βˆ’Μ‡π‘’0(𝑑)π‘‘π‘‘βŸΆ0asπ‘›βŸΆβˆž.(3.25) It is easy to see from the boundedness of {𝑒𝑛} and (3.18) that ξ€œ0π‘˜π‘‡ξ‚€||𝑒𝑛||(𝑑)π‘βˆ’2𝑒𝑛(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0(𝑑)π‘‘π‘‘βŸΆ0asπ‘›βŸΆβˆž.(3.26) Let βˆ«π‘“(𝑒)=(1/𝑝)(0π‘˜π‘‡|𝑒(𝑑)|π‘βˆ«π‘‘π‘‘+0π‘˜π‘‡|̇𝑒(𝑑)|𝑝𝑑𝑑). Then, we have 𝑒𝑓′𝑛,π‘’π‘›βˆ’π‘’0=ξ€œ0π‘˜π‘‡ξ‚€||̇𝑒𝑛||(𝑑)π‘βˆ’2̇𝑒𝑛(𝑑),̇𝑒𝑛(𝑑)βˆ’Μ‡π‘’0+ξ€œ(𝑑)𝑑𝑑0π‘˜π‘‡ξ‚€||𝑒𝑛||(𝑑)π‘βˆ’2𝑒𝑛(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0(𝑑)𝑑𝑑,(3.27)𝑒𝑓′0ξ€Έ,π‘’π‘›βˆ’π‘’0=ξ€œ0π‘˜π‘‡ξ‚€||̇𝑒0||(𝑑)π‘βˆ’2̇𝑒0(𝑑),̇𝑒𝑛(𝑑)βˆ’Μ‡π‘’0+ξ€œ(𝑑)𝑑𝑑0π‘˜π‘‡ξ‚€||𝑒0||(𝑑)π‘βˆ’2𝑒0(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0(𝑑)𝑑𝑑.(3.28) It follows from (3.25) and (3.26) that 𝑒𝑓′𝑛,π‘’π‘›βˆ’π‘’0⟢0asπ‘›βŸΆβˆž.(3.29) From (3.17), we get 𝑒𝑓′0ξ€Έ,π‘’π‘›βˆ’π‘’0⟢0asπ‘›βŸΆβˆž.(3.30) By (3.27), (3.28), and HΓΆlder's inequality, we have ξ«ξ€·π‘’π‘“β€²π‘›ξ€Έξ€·π‘’βˆ’π‘“β€²0ξ€Έ,π‘’π‘›βˆ’π‘’0=ξ€œ0π‘˜π‘‡ξ‚€||̇𝑒𝑛||(𝑑)π‘βˆ’2̇𝑒𝑛(𝑑),̇𝑒𝑛(𝑑)βˆ’Μ‡π‘’0ξ‚ξ€œ(𝑑)𝑑𝑑+0π‘˜π‘‡ξ‚€||𝑒𝑛||(𝑑)π‘βˆ’2𝑒𝑛(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0ξ‚βˆ’ξ€œ(𝑑)𝑑𝑑0π‘˜π‘‡ξ‚€||̇𝑒0(||𝑑)π‘βˆ’2̇𝑒0(𝑑),̇𝑒𝑛(𝑑)βˆ’Μ‡π‘’0(ξ‚ξ€œπ‘‘)π‘‘π‘‘βˆ’0π‘˜π‘‡ξ‚€||𝑒0(||𝑑)π‘βˆ’2𝑒0(𝑑),𝑒𝑛(𝑑)βˆ’π‘’0(=‖‖𝑒𝑑)𝑑𝑑𝑛‖‖𝑝+‖‖𝑒0β€–β€–π‘βˆ’ξ€œ0π‘˜π‘‡ξ‚€||̇𝑒𝑛(||𝑑)π‘βˆ’2̇𝑒𝑛(𝑑),̇𝑒0(ξ‚ξ€œπ‘‘)π‘‘π‘‘βˆ’0π‘˜π‘‡ξ‚€||𝑒𝑛(||𝑑)π‘βˆ’2𝑒𝑛(𝑑),𝑒0(ξ‚βˆ’ξ€œπ‘‘)𝑑𝑑0π‘˜π‘‡ξ‚€||̇𝑒0(||𝑑)π‘βˆ’2̇𝑒0(𝑑),̇𝑒𝑛(ξ‚ξ€œπ‘‘)π‘‘π‘‘βˆ’0π‘˜π‘‡ξ‚€||𝑒0(||𝑑)π‘βˆ’2𝑒0(𝑑),𝑒𝑛(β‰₯‖‖𝑒𝑑)𝑑𝑑𝑛‖‖𝑝+‖‖𝑒0β€–β€–π‘βˆ’ξ‚€β€–β€–π‘’π‘›β€–β€–πΏπ‘βˆ’1𝑝‖‖𝑒0‖‖𝐿𝑝+β€–β€–Μ‡π‘’π‘›β€–β€–πΏπ‘βˆ’1𝑝‖‖̇𝑒0β€–β€–πΏπ‘ξ‚βˆ’ξ‚€β€–β€–π‘’0β€–β€–πΏπ‘βˆ’1𝑝‖‖𝑒𝑛‖‖𝐿𝑝+‖‖̇𝑒0β€–β€–πΏπ‘βˆ’1𝑝‖‖̇𝑒𝑛‖‖𝐿𝑝β‰₯‖‖𝑒𝑛‖‖𝑝+‖‖𝑒0β€–β€–π‘βˆ’ξ€·β€–β€–π‘’π‘›β€–β€–π‘πΏπ‘+‖‖̇𝑒𝑛‖‖𝑝𝐿𝑝(π‘βˆ’1)/𝑝‖‖𝑒0‖‖𝑝𝐿𝑝+‖‖̇𝑒0‖‖𝑝𝐿𝑝1/π‘βˆ’ξ€·β€–β€–π‘’0‖‖𝑝𝐿𝑝+‖‖̇𝑒0‖‖𝑝𝐿𝑝(π‘βˆ’1)/𝑝‖‖𝑒𝑛‖‖𝑝𝐿𝑝+‖‖̇𝑒𝑛‖‖𝑝𝐿𝑝1/𝑝=‖‖𝑒𝑛‖‖𝑝+‖‖𝑒0β€–β€–π‘βˆ’ξ‚€β€–β€–π‘’π‘›β€–β€–π‘βˆ’1‖‖𝑒0β€–β€–+‖‖𝑒0β€–β€–π‘βˆ’1‖‖𝑒𝑛‖‖=ξ‚€β€–β€–π‘’π‘›β€–β€–π‘βˆ’1βˆ’β€–β€–π‘’0β€–β€–π‘βˆ’1ξ‚ξ€·β€–β€–π‘’π‘›β€–β€–βˆ’β€–β€–π‘’0β€–β€–ξ€Έ.(3.31) Hence, from (3.29) and (3.30), we obtain ‖‖𝑒0β‰€π‘›β€–β€–π‘βˆ’1βˆ’β€–β€–π‘’0β€–β€–π‘βˆ’1ξ‚ξ€·β€–β€–π‘’π‘›β€–β€–βˆ’β€–β€–π‘’0β€–β€–ξ€Έβ‰€ξ«π‘“ξ…žξ€·π‘’π‘›ξ€Έβˆ’π‘“ξ…žξ€·π‘’0ξ€Έ,π‘’π‘›βˆ’π‘’0⟢0asπ‘›βŸΆβˆž.(3.32) That is, ‖𝑒𝑛‖→‖𝑒0β€– as π‘›β†’βˆž. Since π‘Š1,π‘π‘˜π‘‡ has the Kadec-Klee property, we have 𝑒𝑛→𝑒0 in π‘Š1,π‘π‘˜π‘‡. Therefore, the functional πœ™π‘˜ satisfies condition (C). Step 2. From (H2), for any small πœ€=πœ€(π‘˜)>0, there exists small enough 𝛿>0 such that 𝐹(𝑑,𝑒)β‰€πœ€|𝑒|𝑝for|𝑒|≀𝛿,a.e.[].π‘‘βˆˆ0,π‘˜π‘‡(3.33) For ξ‚‹π‘Šπ‘’βˆˆ1,π‘π‘˜π‘‡ and ‖𝑒‖𝑝=πœŒπ‘π‘˜=𝛿𝑝/(π‘˜π‘‡)𝑝/π‘ž, it follows from (2.13) that β€–π‘’β€–π‘βˆžβ‰€(π‘˜π‘‡)𝑝/π‘žβ€–Μ‡π‘’β€–π‘πΏπ‘β‰€(π‘˜π‘‡)𝑝/π‘žβ€–π‘’β€–π‘=𝛿𝑝,(3.34) which implies that |𝑒(𝑑)|≀𝛿. Then from (I1), (3.8), and (3.33), we have πœ‘π‘˜1(𝑒)=π‘ξ€œ0π‘˜π‘‡||||̇𝑒(𝑑)𝑝1𝑑𝑑+π‘ξ€œ0π‘˜π‘‡ξ‚€||||𝐿(𝑑)𝑒(𝑑)π‘βˆ’2ξ‚βˆ’ξ€œπ‘’(𝑑),𝑒(𝑑)𝑑𝑑0π‘˜π‘‡πΉ(𝑑,𝑒)𝑑𝑑+π‘˜π‘šξ“π‘—=1𝐼𝑗𝑒𝑑𝑗β‰₯1ξ€Έξ€Έπ‘ξ€œ0π‘˜π‘‡||||̇𝑒(𝑑)𝑝1𝑑𝑑+π‘ξ€œ0π‘˜π‘‡π‘1||||𝑒(𝑑)π‘ξ€œπ‘‘π‘‘βˆ’0π‘˜π‘‡πœ€||||𝑒(𝑑)𝑝1𝑑𝑑β‰₯min𝑝,𝑐1π‘ξ‚Όβ€–π‘’β€–π‘βˆ’π‘˜π‘‡πœ€π›Ώπ‘=𝐢4β€–π‘’β€–π‘βˆ’π‘˜π‘‡πœ€π›Ώπ‘.(3.35) Let πœ€=πœ€(π‘˜)∈(0,𝐢4/2(π‘˜π‘‡)𝑝); then from (3.24), we have πœ‘π‘˜(𝑒)β‰₯𝐢4πœŒπ‘π‘˜βˆ’π‘˜π‘‡πœ€π›Ώπ‘β‰₯𝐢42πœŒπ‘π‘˜β‰‘π›Ό>0(3.36) for all ξ‚‹π‘Šπ‘’βˆˆπ‘‡1,𝑝 and ‖𝑒‖=πœŒπ‘˜. This implies that condition (a) of Lemma 2.2 holds. Step 3. Let 𝑐=max{𝑐𝑗}, π‘—βˆˆπ΅. Choose 𝐢7>(𝑐2/𝑝)+(π‘šπ‘/𝑇); then from (H3), there exists 𝑀4>0 such that 𝐹(𝑑,π‘₯)β‰₯𝐢7|π‘₯|𝑝,|π‘₯|β‰₯𝑀4,a.e.[].π‘‘βˆˆ0,𝑇(3.37) By assumption (A), for |π‘₯|≀𝑀4, there exists 𝐢8=max|π‘₯|≀𝑀4π‘Ž(|π‘₯|)>0 such that ||||𝐹(𝑑,π‘₯)≀𝐢8𝑏(𝑑),a.e.[],π‘‘βˆˆ0,𝑇(3.38) which together with (3.37) implies that 𝐹(𝑑,π‘₯)β‰₯𝐢7|π‘₯|π‘βˆ’πΆ8𝑏(𝑑),βˆ€π‘₯βˆˆβ„π‘,a.e.[].π‘‘βˆˆ0,𝑇(3.39) Thus, from (H1), (I1), (3.8), and (3.39), we have πœ™π‘˜1(𝑒)=π‘ξ€œ0π‘˜π‘‡ξ€·πΏ(𝑑)|𝑒|π‘βˆ’2ξ€Έξ€œπ‘’,π‘’π‘‘π‘‘βˆ’0π‘˜π‘‡πΉ(𝑑,𝑒)𝑑𝑑+π‘˜π‘šξ“π‘—=1𝐼𝑗=π‘˜(𝑒)π‘ξ€œπ‘‡0𝐿(𝑑)|𝑒|π‘βˆ’2ξ€Έξ€œπ‘’,π‘’π‘‘π‘‘βˆ’π‘˜π‘‡0𝐹(𝑑,𝑒)𝑑𝑑+π‘˜π‘šξ“π‘—=1𝐼𝑗≀𝑐(𝑒)2π‘˜π‘ξ€œπ‘‡0|𝑒|π‘ξ€œπ‘‘π‘‘βˆ’π‘˜π‘‡0𝐢7|𝑒|π‘ξ€œπ‘‘π‘‘+π‘˜π‘‡0𝐢8𝑏(𝑑)𝑑𝑑+π‘šπ‘|𝑒|𝑝forπ‘’βˆˆβ„π‘.(3.40) From (H3), we can choose 𝐢7 suitable large such that πœ™π‘˜(𝑒)≀0,βˆ€π‘’βˆˆβ„π‘.(3.41) Let π‘Š1,π‘π‘˜π‘‡=span{π‘’π‘˜}+ℝ𝑁, where π‘’π‘˜=(π‘˜βˆ’1sin(π‘˜βˆ’1πœ”π‘‘)), πœ”=2πœ‹/𝑇. Since π‘Šπ‘‡1,𝑝 is finite dimensional, there exists a constant 𝑑>0 such that ξ‚΅ξ€œπ‘‡0|π‘₯|𝑝𝑑𝑑1/π‘ξ‚΅ξ€œβ‰₯𝑑𝑇0|π‘₯|2𝑑𝑑1/2,βˆ€π‘₯βˆˆπ‘Šπ‘‡1,𝑝.(3.42) By (I1), we have ||πœ“ξ€·π‘’+π‘Ÿπ‘’π‘˜ξ€Έ||=|||||π‘˜π‘šξ“π‘—=1𝐼𝑗𝑒+π‘Ÿπ‘’π‘˜ξ€·π‘‘π‘—|||||β‰€ξ€Έξ€Έπ‘šξ“π‘—=1𝑐𝑗||𝑒+π‘Ÿπ‘’π‘˜ξ€·π‘‘π‘—ξ€Έ||𝑝≀2π‘π‘šπ‘|𝑒|𝑝+2π‘π‘šπ‘π‘Ÿπ‘||π‘’π‘˜ξ€·π‘‘π‘—ξ€Έ||𝑝≀2π‘π‘šπ‘|𝑒|𝑝+2π‘π‘šπ‘π‘Ÿπ‘π‘˜βˆ’π‘β‰€2π‘π‘šπ‘|𝑒|𝑝+2π‘π‘šπ‘π‘Ÿπ‘,π‘’βˆˆβ„π‘.(3.43) From (3.39), (3.42), and (3.43), we obtain πœ™π‘˜ξ€·π‘’+π‘Ÿπ‘’π‘˜ξ€Έ=1π‘ξ€œ0π‘˜π‘‡||π‘ŸΜ‡π‘’π‘˜||(𝑑)π‘ξ€œπ‘‘π‘‘βˆ’0π‘˜π‘‡πΉξ€·π‘‘,𝑒+π‘Ÿπ‘’π‘˜ξ€Έ+1(𝑑)π‘‘π‘‘π‘ξ€œ0π‘˜π‘‡ξ‚€||𝐿(𝑑)𝑒+π‘Ÿπ‘’π‘˜||(𝑑)π‘βˆ’2𝑒+π‘Ÿπ‘’π‘˜ξ€Έ(𝑑),𝑒+π‘Ÿπ‘’π‘˜ξ‚(𝑑)𝑑𝑑+π‘˜π‘šξ“π‘—=1𝐼𝑗𝑒+π‘Ÿπ‘’π‘˜ξ€·π‘‘π‘—β‰€1ξ€Έξ€Έπ‘π‘˜βˆ’2π‘π‘Ÿπ‘πœ”π‘ξ€œ0π‘˜π‘‡||ξ€·π‘˜cosβˆ’1ξ€Έ||πœ”π‘‘π‘π‘π‘‘π‘‘+2π‘ξ€œ0π‘˜π‘‡||𝑒+π‘Ÿπ‘’π‘˜||(𝑑)π‘ξ€œπ‘‘π‘‘+0π‘˜π‘‡πΆ8βˆ’ξ€œπ‘(𝑑)𝑑𝑑0π‘˜π‘‡πΆ7||𝑒+π‘Ÿπ‘’π‘˜||(𝑑)𝑝𝑑𝑑+2π‘π‘šπ‘|𝑒|𝑝+2π‘π‘šπ‘π‘Ÿπ‘β‰€1π‘π‘˜βˆ’2𝑝+1π‘Ÿπ‘πœ”π‘ξ€œπ‘‡0||||cos(πœ”π‘‘)π‘ξ€œπ‘‘π‘‘βˆ’π‘˜π‘‡0𝐢7βˆ’π‘2𝑝||𝑒+π‘Ÿπ‘’1(||𝑑)𝑝+ξ€œπ‘‘π‘‘π‘‡0𝐢8π‘˜π‘(𝑑)𝑑𝑑+2π‘π‘šπ‘|𝑒|𝑝+2π‘π‘šπ‘π‘Ÿπ‘β‰€ξ‚΅π‘‡π‘π‘˜βˆ’2𝑝+1πœ”π‘+2π‘ξ‚Άπ‘Ÿπ‘šπ‘π‘βˆ’π‘˜π‘‘π‘ξ‚΅πΆ7βˆ’π‘2π‘ξ€œξ‚Άξ‚΅π‘‡0||𝑒+π‘Ÿπ‘’1||(𝑑)2𝑑𝑑𝑝/2+2π‘π‘šπ‘|𝑒|𝑝+𝐢9π‘˜β‰€ξ‚΅π‘‡π‘π‘˜βˆ’2𝑝+1πœ”π‘+2π‘ξ‚Άπ‘Ÿπ‘šπ‘π‘βˆ’π‘˜π‘‘π‘ξ‚΅πΆ7βˆ’π‘2π‘ξ€œξ‚Άξ‚΅π‘‡0(|𝑒|2+π‘Ÿ2||𝑒1||(𝑑)2𝑑)𝑝/2+2π‘π‘šπ‘|𝑒|𝑝+𝐢9π‘˜β‰€ξ‚΅π‘‡π‘π‘˜βˆ’2𝑝+1πœ”π‘+2π‘ξ‚Άπ‘Ÿπ‘šπ‘π‘βˆ’π‘˜π‘‘π‘ξ‚΅πΆ7βˆ’π‘2𝑝𝑇|𝑒|2+π‘‡π‘Ÿ22𝑝/2+2π‘π‘šπ‘|𝑒|𝑝+𝐢9π‘˜,βˆ€π‘Ÿβ‰₯0,π‘’βˆˆβ„π‘.(3.44) From (H3), we can choose 𝐢7 suitable such that 𝑑𝑝𝐢7βˆ’π‘2𝑝𝑇2𝑝/2βˆ’23π‘π‘‘π‘šπ‘>0,𝑝𝐢7βˆ’π‘2𝑝𝑇𝑝/2βˆ’2𝑝+1π‘šπ‘>0.(3.45) If π‘˜β‰₯2(𝑇𝑝)1/2π‘πœ”1/2/[𝑑𝑝(𝐢7βˆ’π‘2/𝑝)(𝑇/2)𝑝/2βˆ’23π‘π‘šπ‘]∢=𝐢10, then we get π‘˜βˆ’1πœ™π‘˜ξ€·π‘’+π‘Ÿπ‘’π‘˜ξ€Έβ‰€ξ‚Έπ‘‡π‘π‘˜βˆ’2π‘π‘Ÿπ‘πœ”π‘+2π‘π‘šπ‘π‘˜βˆ’π‘‘π‘ξ‚΅πΆ7βˆ’π‘2𝑝𝑇2𝑝/2ξ‚Ήπ‘Ÿπ‘+𝐢9β‰€ξ‚Έπ‘‡π‘˜βˆ’2π‘πœ”π‘+2π‘π‘šπ‘βˆ’π‘‘π‘ξ‚΅πΆ7βˆ’π‘2𝑝𝑇2𝑝/2ξ‚Ήπ‘Ÿπ‘+𝐢91β‰€βˆ’2𝑑𝑝𝐢7βˆ’π‘2𝑝𝑇2𝑝/2π‘Ÿπ‘+𝐢9,π‘˜βˆ’1πœ™π‘˜ξ€·π‘’+π‘Ÿπ‘’π‘˜ξ€Έ1β‰€βˆ’2𝑑𝑝𝐢7βˆ’π‘2𝑝𝑇𝑝/2|𝑒|𝑝+𝐢9.(3.46) It follows from (3.46) that πœ‘π‘˜ξ€·π‘’+π‘Ÿπ‘’π‘˜ξ€Έβ‰€0,eitherπ‘Ÿβ‰₯π‘Ÿ1or|𝑒|β‰₯π‘Ÿ2,(3.47) where π‘Ÿ1=√2(2𝐢9)1/𝑝/(𝐢7βˆ’π‘2/𝑝)1/𝑝𝑑𝑇1/2, π‘Ÿ2=(2𝐢9)1/𝑝/𝑑(𝐢7βˆ’π‘2/𝑝)1/𝑝𝑇1/2. Notice that for any π‘’βˆˆβ„π‘, we have ‖𝑒‖=‖𝑒‖𝐿𝑝=ξ‚΅ξ€œ0π‘˜π‘‡|𝑒|𝑝𝑑𝑑1/𝑝=(π‘˜π‘‡)1/𝑝𝐢|𝑒|β‰₯10𝑇1/π‘π‘Ÿ2∢=π‘Ÿ3.(3.48) Hence, (3.47) holds for all ‖𝑒‖β‰₯π‘Ÿ3 whenever π‘’βˆˆβ„π‘. Set π‘„π‘˜=ξ‚†π‘Ÿπ‘’π‘˜βˆ£0β‰€π‘Ÿβ‰€π‘Ÿ1,π‘’π‘˜βˆˆξ‚‹π‘Š1,π‘π‘˜π‘‡ξ‚‡βŠ•ξ€½π‘’βˆˆβ„π‘βˆ£β€–π‘’β€–β‰€π‘Ÿ3ξ€Ύ;(3.49) then πœ•π‘„π‘˜=𝑄1π‘˜β‹ƒπ‘„2π‘˜β‹ƒπ‘„3π‘˜, where 𝑄1π‘˜=ξ€½π‘’βˆˆβ„π‘βˆ£β€–π‘’β€–β‰€π‘Ÿ3ξ€Ύ,𝑄2π‘˜=𝑒+π‘Ÿπ‘’π‘˜βˆ£β€–π‘’β€–=π‘Ÿ3ξ€Ί,π‘Ÿβˆˆ0,π‘Ÿ1ξ€»,π‘’π‘˜βˆˆξ‚‹π‘Š1,π‘π‘˜π‘‡ξ‚‡,𝑄3π‘˜=𝑒+π‘Ÿπ‘’π‘˜βˆ£β€–π‘’β€–β‰€π‘Ÿ3,π‘Ÿ=π‘Ÿ1,π‘’π‘˜βˆˆξ‚‹π‘Š1,π‘π‘˜π‘‡ξ‚‡.(3.50) By (3.41) and (3.47), we have πœ‘(𝑒)≀0,π‘’βˆˆπœ•π‘„π‘˜=𝑄1π‘˜ξšπ‘„2π‘˜ξšπ‘„3π‘˜.(3.51) Furthermore, for all 𝑒+π‘Ÿπ‘’π‘˜βˆˆπ‘„π‘˜, it follows from (H1), (3.8), and (3.43) that πœ™π‘˜ξ€·π‘’+π‘Ÿπ‘’π‘˜ξ€Έ=1π‘ξ€œ0π‘˜π‘‡||π‘ŸΜ‡π‘’π‘˜||(𝑑)π‘ξ€œπ‘‘π‘‘βˆ’0π‘˜π‘‡πΉξ€·π‘‘,𝑒+π‘Ÿπ‘’π‘˜ξ€Έ+1(𝑑)π‘‘π‘‘π‘ξ€œ0π‘˜π‘‡ξ‚€||𝐿(𝑑)𝑒+π‘Ÿπ‘’π‘˜||(𝑑)π‘βˆ’2𝑒+π‘Ÿπ‘’π‘˜ξ€Έ(𝑑),𝑒+π‘Ÿπ‘’π‘˜ξ‚(𝑑)𝑑𝑑+π‘˜π‘šξ“π‘—=1𝐼𝑗𝑒+π‘Ÿπ‘’π‘˜ξ€·π‘‘π‘—β‰€1ξ€Έξ€Έπ‘π‘Ÿπ‘ξ€œ0π‘˜π‘‡||Μ‡π‘’π‘˜||(𝑑)𝑝𝑐𝑑𝑑+2π‘ξ€œ0π‘˜π‘‡||𝑒+π‘Ÿπ‘’π‘˜||(𝑑)𝑝𝑑𝑑+2π‘π‘šπ‘|𝑒|𝑝+2π‘π‘šπ‘π‘Ÿπ‘β‰€1π‘π‘˜βˆ’2π‘π‘Ÿπ‘πœ”π‘ξ€œ0π‘˜π‘‡||ξ€·π‘˜cosβˆ’1ξ€Έ||πœ”π‘‘π‘2𝑑𝑑+π‘βˆ’1𝑐2π‘ξ€œ0π‘˜π‘‡ξ€·|𝑒|𝑝+π‘Ÿπ‘π‘˜βˆ’π‘||ξ€·π‘˜sinβˆ’1ξ€Έ||πœ”π‘‘π‘ξ€Έπ‘‘π‘‘+2π‘π‘šπ‘|𝑒|𝑝+2π‘π‘šπ‘π‘Ÿπ‘β‰€1π‘π‘˜βˆ’2𝑝+1π‘Ÿπ‘πœ”π‘ξ€œπ‘‡0||||cos(πœ”π‘‘)𝑝2𝑑𝑑+π‘βˆ’1𝑐2𝑝‖𝑒‖𝑝+π‘Ÿπ‘π‘˜βˆ’π‘+1ξ€œπ‘‡0||||sin(πœ”π‘‘)𝑝+2π‘‘π‘‘π‘π‘šπ‘π‘‡β€–π‘’β€–π‘+2π‘π‘šπ‘π‘Ÿπ‘β‰€π‘‡π‘π‘˜βˆ’2𝑝+1π‘Ÿπ‘πœ”π‘+2π‘βˆ’1𝑐2𝑝‖𝑒‖𝑝+π‘Ÿπ‘π‘˜βˆ’π‘+1𝑇+2π‘π‘šπ‘π‘‡β€–π‘’β€–π‘+2π‘π‘šπ‘π‘Ÿπ‘β‰€π‘‡π‘π‘Ÿπ‘1πœ”π‘+2π‘βˆ’1𝑐2π‘ξ€·π‘Ÿπ‘3+π‘Ÿπ‘1𝑇+2𝑝1π‘šπ‘π‘‡π‘Ÿπ‘3+π‘Ÿπ‘1.(3.52) Then by Lemma 2.2, πœ™π‘˜ has at least a critical point π‘’π‘˜ whose critical value π‘π‘˜ satisfies 0<π›Όβ‰€π‘π‘˜=πœ™π‘˜ξ€·π‘’π‘˜ξ€Έβ‰€π‘‡π‘π‘Ÿπ‘1πœ”π‘+2π‘βˆ’1𝑐2π‘ξ€·π‘Ÿπ‘3+π‘Ÿπ‘1𝑇+2𝑝1π‘šπ‘π‘‡π‘Ÿπ‘3+π‘Ÿπ‘1.(3.53) Similar to the proof of [28], let π‘’π‘˜1 be a π‘˜1𝑇-periodic solution; we can prove that there exists a positive integer π‘˜2>π‘˜1 such that π‘’π‘˜π‘˜1β‰ π‘’π‘˜1 for all π‘˜π‘˜1β‰₯π‘˜2. Otherwise, πœ‘π‘˜(π‘’π‘˜π‘˜1)=π‘˜πœ‘π‘˜(π‘’π‘˜1)β†’βˆž as π‘˜β†’βˆž, which contradicts to (3.53). Repeating this process, we can obtain a sequence {π‘’π‘˜π‘—} of distinct periodic solutions of problem (1.1). From (3.41), we know that π‘’π‘˜π‘— is nonconstant. The proof is complete.

4. Examples

In this section, we give an example to illustrate our result.

Example 4.1. Let 𝑝=3, π‘Ÿ=5, πœ‡=4, and consider the following 𝑝-Laplacian system with impulsive effects 𝑑||||ξ€Έ||||𝑑𝑑̇𝑒(𝑑)̇𝑒(𝑑)βˆ’πΏ(𝑑)𝑒(𝑑)𝑒(𝑑)+βˆ‡πΉ(𝑑,𝑒(𝑑))=0,a.e.Ξ”ξ€·||ξ€·π‘‘π‘‘βˆˆβ„,𝑒(0)βˆ’π‘’(𝑇)=̇𝑒(0)βˆ’Μ‡π‘’(𝑇)=0,̇𝑒𝑖||𝑑̇𝑒𝑖=||𝑑̇𝑒+𝑖||𝑑̇𝑒+π‘–ξ€Έβˆ’||ξ€·π‘‘Μ‡π‘’βˆ’π‘–ξ€Έ||ξ€·π‘‘Μ‡π‘’βˆ’π‘–ξ€Έ=βˆ‡πΌπ‘–ξ€·π‘’ξ€·π‘‘π‘–ξ€Έξ€Έ,𝑖=1,2,…,π‘š.(4.1) Let πΏξ€·ξ€·ξ€·π‘˜(𝑑)=diag1+exp1βˆ’sinβˆ’1ξ€·ξ€·π‘˜πœ”π‘‘ξ€Έξ€Έ,…,1+exp1βˆ’sinβˆ’1,πΌπœ”π‘‘ξ€Έξ€Έξ€Έπ‘–(𝑐π‘₯)=π‘–π‘˜|π‘₯|𝑝,𝐹(𝑑,π‘₯)=1+𝑒3ξ€·ξ€·π‘˜2+sinβˆ’1|πœ”π‘‘ξ€Έξ€Έπ‘₯|5,(4.2) where 𝑐𝑖>0, π‘–βˆˆπ΅. It is easy to check that 𝐹 satisfies (A), (H1), and (H2). By a direct computation, we have lim|π‘₯|β†’βˆžπΉ(𝑑,π‘₯)|π‘₯|3=+∞,limsup|π‘₯|β†’βˆžπΉ(𝑑,π‘₯)|π‘₯|5≀1+𝑒,liminf|π‘₯|β†’βˆž(βˆ‡πΉ(𝑑,π‘₯),π‘₯)βˆ’3𝐹(𝑑,π‘₯)|π‘₯|4β‰₯2(1+𝑒)3,(4.3) which show that (H3), (H4), and (H5) hold. On the other hand, 0≀𝐼𝑖𝑐(π‘₯)β‰€π‘˜|π‘₯|𝑝,βˆ‡πΌπ‘–(π‘₯)π‘₯=π‘π‘π‘–π‘˜|π‘₯|𝑝=𝑝𝐼𝑖(π‘₯),(4.4) where 𝑐=max{𝑐𝑖}, π‘–βˆˆπ΅. It is easy to see that 𝐼𝑖 satisfies (I1) and (I2). Hence, from Theorem 3.1, problem (4.1) has a sequence of distinct nonconstant periodic solutions with period π‘˜π‘—π‘‡ satisfying π‘˜π‘—βˆˆβ„• and π‘˜π‘—β†’βˆž as π‘—β†’βˆž.

Acknowledgments

This paper is partially supported by the NNSF (no. 10771215) of China. W.-Z. Gong is supported by Guangxi Natural Science Foundation (2010GXNSFA013125) and X. H. Tang is supported by the NNSF (no. 10771215) of China.

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