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Journal of Applied Mathematics
Volumeย 2011ย (2011), Article IDย 378389, 19 pages
http://dx.doi.org/10.1155/2011/378389
Research Article

Periodic Solutions for Autonomous (๐‘ž,๐‘)-Laplacian System with Impulsive Effects

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received 17 July 2011; Accepted 31 August 2011

Academic Editor: Yongkunย Li

Copyright ยฉ 2011 Xiaoxia Yang and Haibo Chen. 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 the variational method, some existence theorems are obtained for periodic solutions of autonomous (๐‘ž,๐‘)-Laplacian system with impulsive effects.

1. Introduction

Let ๐ต={1,2,โ€ฆ,๐‘™}, ๐ถ={1,2,โ€ฆ,๐‘˜}, ๐‘™,๐‘˜โˆˆโ„•.

In this paper, we consider the following system: ๐‘‘ฮฆ๐‘‘๐‘ก๐‘ž๎€ทฬ‡๐‘ข1๎€ธ(๐‘ก)=โˆ‡๐‘ข1๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ,(๐‘ก)a.e[],๐‘‘.๐‘กโˆˆ0,๐‘‡ฮฆ๐‘‘๐‘ก๐‘๎€ทฬ‡๐‘ข2๎€ธ(๐‘ก)=โˆ‡๐‘ข2๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ,(๐‘ก)a.e[],๐‘ข.๐‘กโˆˆ0,๐‘‡1(0)โˆ’๐‘ข1(๐‘‡)=ฬ‡๐‘ข1(0)โˆ’ฬ‡๐‘ข1๐‘ข(๐‘‡)=0,2(0)โˆ’๐‘ข2(๐‘‡)=ฬ‡๐‘ข2(0)โˆ’ฬ‡๐‘ข2(๐‘‡)=0,ฮ”ฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ=ฮฆ๐‘ž๎‚€ฬ‡๐‘ข1๎‚€๐‘ก+๐‘—๎‚๎‚โˆ’ฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ท๐‘กโˆ’๐‘—๎€ธ๎€ธ=โˆ‡๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ,๐‘—โˆˆ๐ต,ฮ”ฮฆ๐‘๎€ทฬ‡๐‘ข2๎€ท๐‘ ๐‘š๎€ธ๎€ธ=ฮฆ๐‘๎€ทฬ‡๐‘ข2๎€ท๐‘ +๐‘š๎€ธ๎€ธโˆ’ฮฆ๐‘๎€ทฬ‡๐‘ข2๎€ท๐‘ โˆ’๐‘š๎€ธ๎€ธ=โˆ‡๐พ๐‘š๎€ท๐‘ข2๎€ท๐‘ ๐‘š๎€ธ๎€ธ,๐‘šโˆˆ๐ถ,(1.1) where ๐‘>1,๐‘ž>1, ๐‘‡>0, ๐‘ข(๐‘ก)=(๐‘ข1(๐‘ก),๐‘ข2(๐‘ก))=(๐‘ข11(๐‘ก),๐‘ข21(๐‘ก),โ€ฆ,๐‘ข๐‘1(๐‘ก),๐‘ข12(๐‘ก),๐‘ข22(๐‘ก),โ€ฆ,๐‘ข๐‘2(๐‘ก))๐œ,โ€‰โ€‰๐‘ก๐‘—(๐‘—=1,2,โ€ฆ,๐‘™), and ๐‘ ๐‘š(๐‘š=1,2,โ€ฆ,๐‘˜) are the instants where the impulses occur and 0=๐‘ก0<๐‘ก1<๐‘ก2<โ‹ฏ<๐‘ก๐‘™<๐‘ก๐‘™+1=๐‘‡,0=๐‘ 0<๐‘ 1<๐‘ 2<โ‹ฏ<๐‘ ๐‘˜<๐‘ ๐‘˜+1=๐‘‡, ๐ผ๐‘—โˆถโ„๐‘โ†’โ„(๐‘—โˆˆ๐ต), and ๐พ๐‘šโˆถโ„๐‘โ†’โ„(๐‘šโˆˆ๐ถ) are continuously differentiableฮฆ๐œ‡(๐‘ง)=|๐‘ง|๐œ‡โˆ’2๎ƒฉ๐‘ง=๐‘๎“๐‘–=1๐‘ง2๐‘–๎ƒช(๐œ‡โˆ’2)/2โŽ›โŽœโŽœโŽœโŽœโŽ๐‘ง1โ‹ฎ๐‘ง๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,๐œ‡โˆˆโ„,๐œ‡>1,โˆ‡๐ผ๐‘—(โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐‘ฅ)=๐œ•๐ผ๐‘—๐œ•๐‘ฅ1โ‹ฎ๐œ•๐ผ๐‘—๐œ•๐‘ฅ๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,โˆ‡๐พ๐‘š(โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐‘ฅ)=๐œ•๐พ๐‘š๐œ•๐‘ฅ1โ‹ฎ๐œ•๐พ๐‘š๐œ•๐‘ฅ๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(1.2) and ๐นโˆถโ„๐‘ร—โ„๐‘โ†’โ„ satisfies the following assumption.(A)๐น(๐‘ฅ) is continuously differentiable in (๐‘ฅ1,๐‘ฅ2), and there exist ๐‘Ž1,๐‘Ž2โˆˆ๐ถ(โ„+,โ„+) such that||๐น๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ||โ‰ค๐‘Ž1๎€ท||๐‘ฅ1||๎€ธ+๐‘Ž2๎€ท||๐‘ฅ2||๎€ธ,||๎€ท๐‘ฅโˆ‡๐น1,๐‘ฅ2๎€ธ||โ‰ค๐‘Ž1๎€ท||๐‘ฅ1||๎€ธ+๐‘Ž2๎€ท||๐‘ฅ2||๎€ธ,||๐ผ๐‘—๎€ท๐‘ฅ1๎€ธ||โ‰ค๐‘Ž1๎€ท||๐‘ฅ1||๎€ธ,||โˆ‡๐ผ๐‘—๎€ท๐‘ฅ1๎€ธ||โ‰ค๐‘Ž1๎€ท||๐‘ฅ1||๎€ธ||๐พ,๐‘—โˆˆ๐ต,๐‘š๎€ท๐‘ฅ2๎€ธ||โ‰ค๐‘Ž2๎€ท||๐‘ฅ2||๎€ธ,||โˆ‡๐พ๐‘š๎€ท๐‘ฅ2๎€ธ||โ‰ค๐‘Ž2๎€ท||๐‘ฅ2||๎€ธ,๐‘šโˆˆ๐ถ,(1.3)

for all ๐‘ฅ=(๐‘ฅ1,๐‘ฅ2)โˆˆโ„๐‘ร—โ„๐‘.

When ๐‘=๐‘ž=2, ๐ผ๐‘—โ‰ก0(๐‘—โˆˆ๐ต), ๐พ๐‘šโ‰ก0(๐‘šโˆˆ๐ถ), and ๐น(๐‘ข1,๐‘ข2)=๐น1(๐‘ข1), system (1.1) reduces to the following autonomous second-order Hamiltonian system:ฬˆ๐‘ข1(๐‘ก)=โˆ‡๐‘ข1๐น1๎€ท๐‘ข1๎€ธ,(๐‘ก)a.e[],๐‘ข.๐‘กโˆˆ0,๐‘‡1(0)โˆ’๐‘ข1(๐‘‡)=ฬ‡๐‘ข1(0)โˆ’ฬ‡๐‘ข1(๐‘‡)=0.(1.4) There have been lots of results about the existence of periodic solutions for system (1.4) and nonautonomous second order Hamiltonian systemฬˆ๐‘ข1(๐‘ก)=โˆ‡๐‘ข1๐น1๎€ท๐‘ก,๐‘ข1๎€ธ,(๐‘ก)a.e[],๐‘ข.๐‘กโˆˆ0,๐‘‡1(0)โˆ’๐‘ข1(๐‘‡)=ฬ‡๐‘ข1(0)โˆ’ฬ‡๐‘ข1(๐‘‡)=0,(1.5) (e.g., see [1โ€“21]). Many solvability conditions have been given, for instance, coercive condition, subquadratic condition, superquadratic condition, convex condition, and so on.

When ๐‘=๐‘ž=2, โˆ‡๐ผ๐‘—โ‰ข0(๐‘—โˆˆ๐ต), ๐พ๐‘šโ‰ก0(๐‘šโˆˆ๐ถ), and ๐น(๐‘ข1,๐‘ข2)=๐น1(๐‘ข1), system (1.1) reduces to the following autonomous second-order Hamiltonian system with impulsive effects:ฬˆ๐‘ข1(๐‘ก)=โˆ‡๐‘ข1๐น1๎€ท๐‘ข1๎€ธ,(๐‘ก)a.e[],๐‘ข.๐‘กโˆˆ0,๐‘‡1(0)โˆ’๐‘ข1(๐‘‡)=ฬ‡๐‘ข1(0)โˆ’ฬ‡๐‘ข1(๐‘‡)=0,ฬ‡๐‘ข1๎‚€๐‘ก+๐‘—๎‚โˆ’ฬ‡๐‘ข1๎€ท๐‘กโˆ’๐‘—๎€ธ=โˆ‡๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—.๎€ธ๎€ธ(1.6) Recently, many authors studied the existence of periodic solutions for impulsive differential equations by using variational methods, and lots of interesting results have been obtained. For example, see [22โ€“28]. Especially, nonautonomous second-order Hamiltonian system with impulsive effects is considered in [25, 26] by using the least action principle and the saddle point theorem.

When ๐ผ๐‘—โ‰ก0(๐‘—โˆˆ๐ต) and ๐พ๐‘šโ‰ก0(๐‘šโˆˆ๐ถ), system (1.1) reduces to the following system: ๐‘‘ฮฆ๐‘‘๐‘ก๐‘ž๎€ทฬ‡๐‘ข1๎€ธ(๐‘ก)=โˆ‡๐‘ข1๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ,(๐‘ก)a.e[],๐‘‘.๐‘กโˆˆ0,๐‘‡ฮฆ๐‘‘๐‘ก๐‘๎€ทฬ‡๐‘ข2๎€ธ(๐‘ก)=โˆ‡๐‘ข2๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ,(๐‘ก)a.e[],๐‘ข.๐‘กโˆˆ0,๐‘‡1(0)โˆ’๐‘ข1(๐‘‡)=ฬ‡๐‘ข1(0)โˆ’ฬ‡๐‘ข1๐‘ข(๐‘‡)=0,2(0)โˆ’๐‘ข2(๐‘‡)=ฬ‡๐‘ข2(0)โˆ’ฬ‡๐‘ข2(๐‘‡)=0.(1.7) In [29, 30], PaลŸca and Tang obtained some existence results for system (1.7) by using the least action principle and saddle point theorem. Motivated by [17, 22โ€“30], in this paper, we are concerned with system (1.1) and also use the least action principle and saddle point theorem to study the existence of periodic solution. Our results still improve those in [17] even if system (1.1) reduces to system (1.4).

A function ๐บโˆถโ„๐‘โ†’โ„ is called to be (๐œ†,๐œ‡)-quasiconcave if ๐บ(๐œ†(๐‘ฅ+๐‘ฆ))โ‰ฅ๐œ‡(๐บ(๐‘ฅ)+๐บ(๐‘ฆ)),(1.8) for some ๐œ†,๐œ‡>0 and ๐‘ฅ,๐‘ฆโˆˆโ„๐‘.

Next, we state our main results.

Theorem 1.1. Let ๐‘ž๎…ž and ๐‘๎…ž be such that 1/๐‘ž+1/๐‘ž๎…ž=1 and 1/๐‘+1/๐‘๎…ž=1. Suppose ๐น satisfies assumption (A) and the following conditions: (F1)there exist 0<๐‘Ÿ1<๎€ท๐‘ž๎…ž๎€ธ+1๐‘ž/๐‘žโ€ฒ๐‘‡๐‘žฮ˜(๐‘ž,๐‘ž๎…ž),0<๐‘Ÿ2<๎€ท๐‘๎…ž๎€ธ+1๐‘/๐‘โ€ฒ๐‘‡๐‘ฮ˜(๐‘,๐‘๎…ž),(1.9) such that ๎€ทโˆ‡๐‘ฅ1๐น๎€ท๐‘ฅ1,๐‘ฅ2๎€ธโˆ’โˆ‡๐‘ฆ1๐น๎€ท๐‘ฆ1,๐‘ฆ2๎€ธ,๐‘ฅ1โˆ’๐‘ฆ1๎€ธโ‰ฅโˆ’๐‘Ÿ1||x1โˆ’๐‘ฆ1||๐‘ž๎€ท๐‘ฅ,โˆ€1,๐‘ฅ2๎€ธ,๎€ท๐‘ฆ1,๐‘ฆ2๎€ธโˆˆโ„๐‘ร—โ„๐‘,๎€ทโˆ‡๐‘ฅ2๐น๎€ท๐‘ฅ1,๐‘ฅ2๎€ธโˆ’โˆ‡๐‘ฆ2๐น๎€ท๐‘ฆ1,๐‘ฆ2๎€ธ,๐‘ฅ2โˆ’๐‘ฆ2๎€ธโ‰ฅโˆ’๐‘Ÿ2||๐‘ฅ2โˆ’๐‘ฆ2||๐‘๎€ท๐‘ฅ,โˆ€1,๐‘ฅ2๎€ธ,๎€ท๐‘ฆ1,๐‘ฆ2๎€ธโˆˆโ„๐‘ร—โ„๐‘,(1.10) where ฮ˜๎€ท๐‘ž,๐‘ž๎…ž๎€ธ=๎€œ10๎‚ƒ๐‘ ๐‘žโ€ฒ+1+(1โˆ’๐‘ )๐‘žโ€ฒ+1๎‚„๐‘ž/๐‘žโ€ฒฮ˜๎€ท๐‘‘๐‘ ,๐‘,๐‘๎…ž๎€ธ=๎€œ10๎‚ƒ๐‘ ๐‘โ€ฒ+1+(1โˆ’๐‘ )๐‘โ€ฒ+1๎‚„๐‘/๐‘โ€ฒ๐‘‘๐‘ ,(1.11)(F2)๐น(๐‘ฅ)โ†’+โˆž,as|๐‘ฅ|โ†’โˆž,where๐‘ฅ=(๐‘ฅ1,๐‘ฅ2), (I1) there exists ๐›ฝโˆˆโ„ such that ๐ผ๐‘—(๐‘ฅ)โ‰ฅ๐›ฝ,โˆ€๐‘ฅโˆˆโ„๐‘๐พ,๐‘—โˆˆ๐ต,๐‘š(๐‘ฅ)โ‰ฅ๐›ฝ,โˆ€๐‘ฅโˆˆโ„๐‘,๐‘šโˆˆ๐ถ.(1.12)Then, system (1.1) has at least one solution in ๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘, where ๐‘Š๐‘‡1,๐‘ ={๐‘ขโˆถ[0,๐‘‡]โ†’โ„๐‘โˆฃ๐‘ข is absolutely continuous,๐‘ข(0)=๐‘ข(๐‘‡) andฬ‡๐‘ขโˆˆ๐ฟ๐‘ (0,๐‘‡;โ„๐‘)},๐‘ โˆˆโ„.
Furthermore, if ๐ผ๐‘—โ‰ก0(๐‘—โˆˆ๐ต), ๐พ๐‘šโ‰ก0(๐‘šโˆˆ๐ถ) and the following condition holds:
(F3) there exist ๐›ฟ>0, ๐‘Žโˆˆ[0,(๐‘žโ€ฒ+1)๐‘ž/๐‘žโ€ฒ/๐‘ž๐‘‡๐‘žฮ˜(๐‘ž,๐‘ž๎…ž)) and ๐‘โˆˆ[0,(๐‘๎…ž+1)๐‘/๐‘โ€ฒ/(๐‘๐‘‡๐‘ฮ˜(p,๐‘๎…ž))) such that ||๐‘ฅโˆ’๐‘Ž1||๐‘ž||๐‘ฅโˆ’๐‘2||๐‘๎€ท๐‘ฅโ‰ค๐น1,๐‘ฅ2๎€ธ||๐‘ฅโ‰ค0,โˆ€1||||๐‘ฅโ‰ค๐›ฟ,2||โ‰ค๐›ฟ,(1.13) then system (1.7) has at least two nonzero solutions in ๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘.

When ๐‘=๐‘ž=2, ๐น(๐‘ฅ1,๐‘ฅ2)=๐น1(๐‘ฅ1), by Theorem 1.1, it is easy to get the following corollary.

Corollary 1.2. Suppose ๐น1 satisfies the following conditions: (A)๎…ž๐น1(๐‘ง) is continuously differentiable in ๐‘ง and there exists ๐‘Ž1โˆˆ๐ถ(โ„+,โ„+) such that ||๐น1||(๐‘ง)โ‰ค๐‘Ž1||(|๐‘ง|),โˆ‡๐น1||(๐‘ง)โ‰ค๐‘Ž1||๐ผ(|๐‘ง|),๐‘—||(๐‘ง)โ‰ค๐‘Ž1||(|๐‘ง|),โˆ‡๐ผ๐‘—||(๐‘ง)โ‰ค๐‘Ž1(|๐‘ง|),๐‘—โˆˆ๐ต,(1.14) for all ๐‘งโˆˆโ„๐‘. (F1)๎…ž there exists 0<๐‘Ÿ<6/๐‘‡2 such that ๎€ทโˆ‡๐‘ง๐น1(๐‘ง)โˆ’โˆ‡๐‘ค๐น1๎€ธ(๐‘ค),๐‘งโˆ’๐‘คโ‰ฅโˆ’๐‘Ÿ|๐‘งโˆ’๐‘ค|2,โˆ€๐‘ง,๐‘คโˆˆโ„๐‘,(1.15)(F2)๎…ž๐น1(๐‘ง)โ†’+โˆž,as|๐‘ง|โ†’โˆž,๐‘งโˆˆโ„๐‘; (I1)๎…žthere exists ๐›ฝโˆˆโ„ such that ๐ผ๐‘—(๐‘ง)โ‰ฅ๐›ฝ,โˆ€๐‘งโˆˆโ„๐‘,๐‘—โˆˆ๐ต.(1.16)Then, system (1.6) has at least one solution in ๐‘Š๐‘‡1,2. Furthermore, if ๐ผ๐‘—โ‰ก0(๐‘—โˆˆ๐ต) and the following condition holds:(F3)๎…ž there exist ๐›ฟ>0 and ๐‘Žโˆˆ[0,(3/๐‘‡2)) such that โˆ’๐‘Ž|๐‘ง|2โ‰ค๐น1(๐‘ง)โ‰ค0,โˆ€๐‘งโˆˆโ„๐‘,|๐‘ง|โ‰ค๐›ฟ,(1.17) then system (1.4) has at least two nonzero solutions in ๐‘Š๐‘‡1,2.

For the Sobolev space ๎‚‹๐‘Š๐‘‡1,2, one has the following sharp estimates (see in [3, Propositionโ€‰โ€‰1.2]): ๎€œ๐‘‡0||||๐‘ข(๐‘ก)2๐‘‡๐‘‘๐‘กโ‰ค24๐œ‹2๎€œ๐‘‡0||||ฬ‡๐‘ข(๐‘ก)2๎€ท๐‘‘๐‘กWirtinger๎…žsinequality๎€ธ,(1.18)โ€–๐‘ขโ€–2โˆžโ‰ค๐‘‡๎€œ12๐‘‡0||||ฬ‡๐‘ข(๐‘ก)2๎€ท๐‘‘๐‘กSobolev๎…žsinequality๎€ธ.(1.19) By the above two inequalities, we can obtain the following better results than by Corollary 1.2.

Theorem 1.3. Suppose ๐น1 satisfies assumption (A)โ€ฒ, (F2)โ€ฒ, (I1)โ€ฒ and (F1)๎…ž๎…ž there exists 0<๐‘Ÿ<4๐œ‹2/๐‘‡2 such that (1.15) holds.Then, system (1.6) has at least one solution in ๐‘Š๐‘‡1,2. Furthermore, if ๐ผ๐‘—โ‰ก0(๐‘—โˆˆ๐ต) and the following condition holds:(F3)๎…ž๎…ž there exist ๐›ฟ>0 and ๐‘Žโˆˆ[0,(2๐œ‹2)/๐‘‡2) such that โˆ’๐‘Ž|๐‘ง|2โ‰ค๐น1(๐‘ง)โ‰ค0,โˆ€๐‘งโˆˆโ„๐‘,|๐‘ง|โ‰ค๐›ฟ,(1.20) then system (1.4) has at least two nonzero solutions in ๐‘Š๐‘‡1,2.

Moreover, for system (1.6), we have the following additional result.

Theorem 1.4. Suppose ๐น1 satisfies assumption (A)โ€ฒ, (F1)โ€ฒ โ€ฒ and the following conditions: (F4)๐น1(๐‘ง) is (๐œ†,๐œ‡)-quasiconcave on โ„๐‘,(F5)๐น1(๐‘ง)โ†’โˆ’โˆž as |๐‘ง|โ†’+โˆž, ๐‘งโˆˆโ„๐‘,(I2) there exist ๐‘‘๐‘—>0(๐‘—โˆˆ๐ต) such that ||โˆ‡๐ผ๐‘—||(๐‘ง)โ‰ค๐‘‘๐‘—,โˆ€๐‘งโˆˆโ„๐‘,๐‘—โˆˆ๐ต,(1.21)(I3) there exist ๐‘๐‘—>0,๐‘๐‘—>0, ๐›พ๐‘—โˆˆโ„, ๐›ผ๐‘—โˆˆ[0,2)(๐‘—โˆˆ๐ต) such that โˆ’๐‘๐‘—|z|๐›ผ๐‘—โˆ’๐‘๐‘—โ‰ค๐ผ๐‘—(๐‘ง)โ‰ค๐›พ๐‘—,โˆ€๐‘งโˆˆโ„๐‘,๐‘—โˆˆ๐ต.(1.22)Then, system (1.6) has at least one solution in ๐‘Š๐‘‡1,2.

Remark 1.5. In [17], Yang considered the second-order Hamiltonian system with no impulsive effects, that is, system (1.4). When ๐ผ๐‘—โ‰ก0(๐‘—โˆˆ๐ต), our Theorems 1.3 and 1.4 still improve those results in [17]. To be precise, the restriction of ๐‘Ÿ is relaxed, and some unnecessary conditions in [17] are deleted. In [17], the restriction of ๐‘Ÿ is 0<๐‘Ÿ<๐‘‡/12, which is not right. In fact, from his proof, it is easy to see that it should be 0<๐‘Ÿ<12/๐‘‡2. Obviously, our restriction 0<๐‘Ÿ<4๐œ‹2/๐‘‡2 is better. Moreover, in our Theorem 1.4, we delete such conditions (of in [17, Theoremโ€‰โ€‰1]): โˆ‡๐น1(0)=0, and there exist positive constants ๐‘€,๐‘ such that ๐น1(๐‘ง)โ‰ฅโˆ’๐‘€|๐‘ง|2โˆ’๐‘,๐‘งโˆˆโ„๐‘.(1.23) Finally, it is remarkable that Theorems 1.3 and 1.4 are also different from those results in [1โ€“16]. We can find an example satisfying our Theorem 1.3 but not satisfying the results in [1โ€“21]. For example, let ๐น1๐œ‹(๐‘ง)=22๐‘‡2๎‚€||๐‘ง1||4+||๐‘ง2||4||๐‘ง+โ‹ฏ+๐‘||4๎‚โˆ’๐œ‹24๐‘‡2|๐‘ง|2,(1.24) where ๐‘ง=(๐‘ง1,โ€ฆ,๐‘ง๐‘)๐œ. We can also find an example satisfying our Theorem 1.4 but not satisfying the results in [1โ€“21]. For example, let ๐น1๐‘Ÿ(๐‘ง)=โˆ’2|๐‘ง|2,(1.25) where 12/๐‘‡2<๐‘Ÿ<4๐œ‹2/๐‘‡2.

2. Variational Structure and Some Preliminaries

The norm in ๐‘Š๐‘‡1,๐‘ is defined byโ€–๐‘ขโ€–๐‘Š๐‘‡1,๐‘=๎‚ธ๎€œ๐‘‡0||||๐‘ข(๐‘ก)๐‘๎€œ๐‘‘๐‘ก+๐‘‡0||||ฬ‡๐‘ข(๐‘ก)๐‘๎‚น๐‘‘๐‘ก1/๐‘.(2.1) Setโ€–๐‘ขโ€–๐‘=๎‚ต๎€œ๐‘‡0||||๐‘ข(๐‘ก)๐‘๎‚ถ๐‘‘๐‘ก1/๐‘,โ€–๐‘ขโ€–โˆž=max๐‘กโˆˆ[0,๐‘‡]||||.๐‘ข(๐‘ก)(2.2) Let๎‚‹๐‘Š๐‘‡1,๐‘=๎‚ป๐‘ขโˆˆ๐‘Š๐‘‡1,๐‘โˆฃ๎€œ๐‘‡0๎‚ผ.๐‘ข(๐‘ก)๐‘‘๐‘ก=0(2.3) Obviously, ๐‘Š๐‘‡1,๐‘ is a reflexive Banach space. It is easy to know that ๎‚‹๐‘Š๐‘‡1,๐‘ is a subset of ๐‘Š๐‘‡1,๐‘ and ๐‘Š๐‘‡1,๐‘=โ„๐‘โŠ•๎‚‹๐‘Š๐‘‡1,๐‘. In this paper, we will use the space ๐‘Š defined by ๐‘Š=๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘๎€ท๐‘ข,๐‘ข(๐‘ก)=1(๐‘ก),๐‘ข2๎€ธ,(๐‘ก)(2.4) with the norm โ€–(๐‘ข1,๐‘ข2)โ€–๐‘Š=โ€–๐‘ข1โ€–๐‘Š๐‘‡1,๐‘ž+โ€–๐‘ข2โ€–๐‘Š๐‘‡1,๐‘. It is clear that ๐‘Š is a reflexive Banach space. Let ๎‚‹๎‚‹๐‘Š๐‘Š=๐‘‡1,๐‘žร—๎‚‹๐‘Š๐‘‡1,๐‘. Then, ๎‚‹๐‘Š๐‘Š=(๐‘‡1,๐‘žร—๎‚‹๐‘Š๐‘‡1,๐‘)โŠ•(โ„๐‘ร—โ„๐‘).

Lemma 2.1 (see [31] or [32]). Each ๐‘ขโˆˆ๐‘Š๐‘‡1,๐‘ and each ๐‘ฃโˆˆ๐‘Š๐‘‡1,๐‘ž can be written as ๐‘ข(๐‘ก)=๐‘ข+ฬƒ๐‘ข(๐‘ก) and ๐‘ฃ(๐‘ก)=ฬƒ๐‘ฃ+๐‘ฃ(๐‘ก) with 1๐‘ข=๐‘‡๎€œ๐‘‡0๎€œ๐‘ข(๐‘ก)๐‘‘๐‘ก,๐‘‡0ฬƒ๐‘ข(๐‘ก)๐‘‘๐‘ก=0,1๐‘ฃ=๐‘‡๎€œ๐‘‡0๎€œ๐‘ฃ(๐‘ก)๐‘‘๐‘ก,๐‘‡0ฬƒ๐‘ฃ(๐‘ก)๐‘‘๐‘ก=0.(2.5) Then, โ€–ฬƒ๐‘ขโ€–โˆžโ‰ค๎‚ต๐‘‡๎‚ถ๐‘โ€ฒ+11/๐‘๎…ž๎‚ต๎€œ๐‘‡0||||ฬ‡๐‘ข(๐‘ )๐‘๎‚ถ๐‘‘๐‘ 1/๐‘ฬƒ,โ€–๐‘ฃโ€–โˆžโ‰ค๎‚ต๐‘‡๎‚ถ๐‘žโ€ฒ+11/๐‘ž๎…ž๎‚ต๎€œ๐‘‡0||ฬ‡||๐‘ฃ(๐‘ )๐‘ž๎‚ถ๐‘‘๐‘ 1/๐‘ž,(2.6)๎€œ๐‘‡0||||ฬƒ๐‘ข(๐‘ )๐‘๐‘‡๐‘‘๐‘ โ‰ค๐‘ฮ˜๎€ท๐‘,๐‘๎…ž๎€ธ(๐‘๎…ž+1)๐‘/๐‘โ€ฒ๎€œ๐‘‡0||||ฬ‡๐‘ข(๐‘ )๐‘๎€œ๐‘‘๐‘ ,๐‘‡0||ฬƒ||๐‘ฃ(๐‘ )๐‘ž๐‘‡๐‘‘๐‘ โ‰ค๐‘žฮ˜๎€ท๐‘ž,๐‘ž๎…ž๎€ธ(๐‘žโ€ฒ+1)๐‘ž/๐‘žโ€ฒ๎€œ๐‘‡0||ฬ‡||๐‘ฃ(๐‘ )๐‘ž๐‘‘๐‘ ,(2.7) where ฮ˜๎€ท๐‘,๐‘๎…ž๎€ธ=๎€œ10๎‚ƒ๐‘ ๐‘โ€ฒ+1+(1โˆ’๐‘ )๐‘โ€ฒ+1๎‚„๐‘/๐‘โ€ฒ๎€ท๐‘‘๐‘ ,ฮ˜๐‘ž,๐‘ž๎…ž๎€ธ=๎€œ10๎€บ๐‘ ๐‘ž๎…ž+1+(1โˆ’๐‘ )๐‘ž๎…ž+1๎€ป๐‘ž/๐‘žโ€ฒ๐‘‘๐‘ .(2.8)

Note that if ๐‘ขโˆˆ๐‘Š๐‘‡1,๐‘, then ๐‘ข is absolutely continuous. However, we cannot guarantee that ฬ‡๐‘ข is also continuous. Hence, it is possible that ฮ”ฮฆ๐‘(ฬ‡๐‘ข(๐‘ก))=ฮฆ๐‘(ฬ‡๐‘ข(๐‘ก+))โˆ’ฮฆ๐‘(ฬ‡๐‘ข(๐‘กโˆ’))โ‰ 0, which results in impulsive effects.

Following the idea in [22], one takes ๐‘ฃ1โˆˆ๐‘Š๐‘‡1,๐‘ž and multiplies the two sides of๐‘‘๎‚€||๐‘‘๐‘กฬ‡๐‘ข1||(๐‘ก)๐‘žโˆ’2ฬ‡๐‘ข1๎‚(๐‘ก)โˆ’โˆ‡๐‘ฅ1๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ(๐‘ก)=0,(2.9)

by ๐‘ฃ1 and integrate from 0 to ๐‘‡, one obtains๎€œ๐‘‡0๎‚ƒ๐‘‘๎‚€||๐‘‘๐‘กฬ‡๐‘ข1||(๐‘ก)๐‘žโˆ’2ฬ‡๐‘ข1๎‚(๐‘ก)โˆ’โˆ‡๐‘ฅ1๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ๎‚„๐‘ฃ(๐‘ก)1.(๐‘ก)๐‘‘๐‘ก=0(2.10) Note that ๐‘ฃ1(๐‘ก) is continuous. So, ๐‘ฃ1(๐‘กโˆ’๐‘—)=๐‘ฃ1(๐‘ก+๐‘—)=๐‘ฃ1(๐‘ก๐‘—). Combining ฬ‡๐‘ข1(0)โˆ’ฬ‡๐‘ข1(๐‘‡)=0, one has ๎€œ๐‘‡0๎ƒฉ๐‘‘ฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ธ(๐‘ก)๐‘‘๐‘ก,๐‘ฃ1๎ƒช(๐‘ก)๐‘‘๐‘ก=๐‘™๎“๐‘—=0๎€œ๐‘ก๐‘—+1๐‘ก๐‘—๎ƒฉ๐‘‘๎€ทฮฆ๐‘ž๎€ทฬ‡๐‘ข1(๐‘ก)๎€ธ๎€ธ๐‘‘๐‘ก,๐‘ฃ1๎ƒช=(๐‘ก)๐‘‘๐‘ก๐‘™๎“๐‘—=0ฮฆ๎‚ƒ๎‚€๐‘ž๎‚€ฬ‡๐‘ข1๎‚€๐‘กโˆ’๐‘—+1๎‚๎‚,๐‘ฃ1๎‚€๐‘กโˆ’๐‘—+1โˆ’๎‚€ฮฆ๎‚๎‚๐‘ž๎‚€ฬ‡๐‘ข1๎‚€๐‘ก+๐‘—๎‚๎‚,๐‘ฃ1๎‚€๐‘ก+๐‘—โˆ’๎‚๎‚๎‚„๐‘‘๐‘ก๐‘™๎“๐‘—=0๎€œ๐‘ก๐‘—+1๐‘ก๐‘—๎€ทฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ธ,ฬ‡๐‘ฃ(๐‘ก)1๎€ธ=๎€ทฮฆ(๐‘ก)๐‘‘๐‘ก๐‘ž๎€ทฬ‡๐‘ข1๎€ธ(๐‘‡),๐‘ฃ1๎€ธโˆ’๎€ทฮฆ(๐‘‡)๐‘ž๎€ทฬ‡๐‘ข1๎€ธ(0),๐‘ฃ1๎€ธโˆ’(0)๐‘™๎“๐‘—=1๎€ทฮ”ฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ,๐‘ฃ1๎€ท๐‘ก๐‘—โˆ’๎€œ๎€ธ๎€ธ๐‘‡0๎€ทฮฆ๐‘ž๎€ทฬ‡๐‘ข1(๎€ธ,ฬ‡๐‘ฃ๐‘ก)1(๎€ธ๐‘ก)๐‘‘๐‘ก=โˆ’๐‘™๎“๐‘—=1๎€ทโˆ‡๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ,๐‘ฃ1๎€ท๐‘ก๐‘—โˆ’๎€œ๎€ธ๎€ธ๐‘‡0๎€ทฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ธ,ฬ‡๐‘ฃ(๐‘ก)1๎€ธ(๐‘ก)๐‘‘๐‘ก.(2.11) Combining with (2.10), one has๎€œ๐‘‡0๎€ทฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ธ,ฬ‡๐‘ฃ(๐‘ก)1๎€ธ(๐‘ก)๐‘‘๐‘ก+๐‘™๎“๐‘—=1๎€ทโˆ‡๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ,๐‘ฃ1๎€ท๐‘ก๐‘—+๎€œ๎€ธ๎€ธ๐‘‡0๎€ทโˆ‡๐‘ฅ1๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ(๐‘ก),๐‘ฃ1๎€ธ(๐‘ก)๐‘‘๐‘ก=0.(2.12) Similarly, one can get๎€œ๐‘‡0๎€ทฮฆ๐‘๎€ทฬ‡๐‘ข2๎€ธ,ฬ‡๐‘ฃ(๐‘ก)2๎€ธ(๐‘ก)๐‘‘๐‘ก+๐‘˜๎“๐‘š=1๎€ทโˆ‡๐พ๐‘š๎€ท๐‘ข2๎€ท๐‘ ๐‘š๎€ธ๎€ธ,๐‘ฃ2๎€ท๐‘ ๐‘š+๎€œ๎€ธ๎€ธ๐‘‡0๎€ทโˆ‡๐‘ฅ2๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ(๐‘ก),๐‘ฃ2๎€ธ(๐‘ก)๐‘‘๐‘ก=0,(2.13) for all ๐‘ฃ2โˆˆ๐‘Š๐‘‡1,๐‘. Considering the above equalities, one introduces the following concept of the weak solution for system (1.1).

Definition 2.2. We say that a function ๐‘ข=(๐‘ข1,๐‘ข2)โˆˆ๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘ is a weak solution of system (1.1) if๎€œ๐‘‡0๎€ทฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ธ,ฬ‡๐‘ฃ(๐‘ก)1๎€ธ(๐‘ก)๐‘‘๐‘ก+๐‘™๎“๐‘—=1๎€ทโˆ‡๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ,๐‘ฃ1๎€ท๐‘ก๐‘—๎€œ๎€ธ๎€ธ=โˆ’๐‘‡0๎€ทโˆ‡๐‘ฅ1๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ(๐‘ก),๐‘ฃ1๎€ธ๎€œ(๐‘ก)๐‘‘๐‘ก,๐‘‡0๎€ทฮฆ๐‘๎€ทฬ‡๐‘ข2๎€ธ,ฬ‡๐‘ฃ(๐‘ก)2๎€ธ(๐‘ก)๐‘‘๐‘ก+๐‘˜๎“๐‘š=1๎€ทโˆ‡๐พ๐‘š๎€ท๐‘ข2๎€ท๐‘ ๐‘š๎€ธ๎€ธ,๐‘ฃ2๎€ท๐‘ ๐‘š๎€œ๎€ธ๎€ธ=โˆ’๐‘‡0๎€ทโˆ‡๐‘ฅ2๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ(๐‘ก),๐‘ฃ2๎€ธ(๐‘ก)๐‘‘๐‘ก(2.14) holds for any ๐‘ฃ=(๐‘ฃ1,๐‘ฃ2)โˆˆ๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘.
Define the functional ๐œ‘โˆถ๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘โ†’โ„ by ๐œ‘๎€ท๐‘ข1,๐‘ข2๎€ธ=1๐‘ž๎€œ๐‘‡0||ฬ‡๐‘ข1(t)||๐‘ž1๐‘‘๐‘ก+๐‘๎€œ๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘๎€œ๐‘‘๐‘ก+๐‘‡0๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ+(๐‘ก)๐‘‘๐‘ก๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—+๎€ธ๎€ธ๐‘˜๎“๐‘š=1๐พ๐‘š๎€ท๐‘ข2๎€ท๐‘ ๐‘š๎€ท๐‘ข๎€ธ๎€ธ=๐œ™1,๐‘ข2๎€ธ๎€ท๐‘ข+๐œ“1,๐‘ข2๎€ธ,(2.15) where (๐‘ข1,๐‘ข2)โˆˆ๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘, ๐œ™๎€ท๐‘ข1,๐‘ข2๎€ธ=1๐‘ž๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)๐‘ž1๐‘‘๐‘ก+๐‘๎€œ๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘๎€œ๐‘‘๐‘ก+๐‘‡0๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ๐œ“๎€ท๐‘ข(๐‘ก)๐‘‘๐‘ก,1,๐‘ข2๎€ธ=๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—+๎€ธ๎€ธ๐‘˜๎“๐‘š=1๐พ๐‘š๎€ท๐‘ข2๎€ท๐‘ ๐‘š.๎€ธ๎€ธ(2.16) By assumption (A) and [33], we know that ๐œ™โˆˆ๐ถ1(๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘,โ„). The continuity of ๐ผ๐‘—(๐‘—โˆˆ๐ต) and ๐พ๐‘š(๐‘šโˆˆ๐ถ) implies that ๐œ“โˆˆ๐ถ1(๐‘Š๐‘‡1,๐‘ร—๐‘Š๐‘‡1,๐‘,โ„). So, ๐œ‘โˆˆ๐ถ1(๐‘Š๐‘‡1,๐‘,โ„), and for all (๐‘ฃ1,๐‘ฃ2)โˆˆ๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘, we have ๎ซ๐œ‘๎…ž๎€ท๐‘ข1,๐‘ข2๎€ธ,๎€ท๐‘ฃ1,๐‘ฃ2=๎€œ๎€ธ๎ฌ๐‘‡0๎€ทฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ธ,ฬ‡๐‘ฃ(๐‘ก)1๎€ธ๎€œ(๐‘ก)๐‘‘๐‘ก+๐‘‡0๎€ทฮฆ๐‘๎€ทฬ‡๐‘ข2๎€ธ,ฬ‡๐‘ฃ(๐‘ก)2๎€ธ+๎€œ(๐‘ก)๐‘‘๐‘ก๐‘‡0๎€ทโˆ‡๐‘ฅ1๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ(๐‘ก),๐‘ฃ1๎€ธ๎€œ(๐‘ก)๐‘‘๐‘ก+๐‘‡0๎€ทโˆ‡๐‘ฅ2๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ(๐‘ก),๐‘ฃ2๎€ธ+(๐‘ก)๐‘‘๐‘ก๐‘™๎“๐‘—=1๎€ทโˆ‡๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ,๐‘ฃ1๎€ท๐‘ก๐‘—+๎€ธ๎€ธ๐‘˜๎“๐‘š=1๎€ทโˆ‡๐พ๐‘š๎€ท๐‘ข2๎€ท๐‘ ๐‘š๎€ธ๎€ธ,๐‘ฃ2๎€ท๐‘ ๐‘š.๎€ธ๎€ธ(2.17) Definition 2.2 shows that the critical points of ๐œ‘ correspond to the weak solutions of system (1.1).
We will use the following lemma to seek the critical point of ๐œ‘.

Lemma 2.3 (see [3, Theoremโ€‰โ€‰1.1]). If ๐œ‘ is weakly lower semicontinuous on a reflexive Banach space ๐‘‹ and has a bounded minimizing sequence, then ๐œ‘ has a minimum on ๐‘‹.

Lemma 2.4 (see [34]). Let ๐œ‘ be a ๐ถ1 function on ๐‘‹=๐‘‹1โŠ•๐‘‹2 with ๐œ‘(0)=0, satisfying (PS) condition, and assume that for some ๐œŒ>0, ๐œ‘(๐‘ข)โ‰ฅ0,for๐‘ขโˆˆ๐‘‹1๐œ‘,โ€–๐‘ขโ€–โ‰ค๐œŒ,(๐‘ข)โ‰ค0,for๐‘ขโˆˆ๐‘‹2,โ€–๐‘ขโ€–โ‰ค๐œŒ.(2.18) Assume also that ๐œ‘ is bounded below and inf๐‘‹๐œ‘<0, then ๐œ‘ has at least two nonzero critical points.

Lemma 2.5 (see [35, Theoremโ€‰โ€‰4.6]). Let ๐‘‹=๐‘‹1โŠ•๐‘‹2, where ๐‘‹ is a real Banach space and ๐‘‹1โ‰ {0} and is finite dimensional. Suppose that ๐œ‘โˆˆ๐ถ1(๐‘‹,โ„) satisfies (PS)-condition and (๐œ‘1)there is a constant ๐›ผ and a bounded neighborhood ๐ท of 0 in ๐‘‹1 such that ๐œ‘โˆฃ๐œ•๐ทโ‰ค๐›ผ,(๐œ‘2) there is a constant ๐›ฝ>๐›ผ such that ๐œ‘โˆฃ๐‘‹2โ‰ฅ๐›ฝ.
Then, ๐œ‘ possesses a critical value ๐‘โ‰ฅ๐›ฝ. Moreover, ๐‘ can be characterized as ๐‘=infโ„Žโˆˆฮ“max๐‘ขโˆˆ๐ท๐œ‘(โ„Ž(๐‘ข)),(2.19) where, ๎‚†๎‚€ฮ“=โ„Žโˆˆ๐ถ๎‚๎‚‡.๐ท,๐‘‹โˆฃโ„Ž=๐‘–๐‘‘on๐œ•๐ท(2.20)

3. Proof of Theorems

Lemma 3.1. Under assumption (A), ๐œ‘ is weakly lower semicontinuous on ๐‘Š๐‘‡1,๐‘žร—๐‘Š๐‘‡1,๐‘.

Proof. Let ๐œ™1๎€ท๐‘ข1,๐‘ข2๎€ธ=1๐‘ž๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)๐‘ž1๐‘‘๐‘ก+๐‘๎€œ๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘๐œ™๐‘‘๐‘ก,2๎€ท๐‘ข1,๐‘ข2๎€ธ=๎€œ๐‘‡0๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ(๐‘ก)๐‘‘๐‘ก.(3.1) Since ๐œ™1๎‚ต๐‘ข1+๐‘ฃ12,๐‘ข2+๐‘ฃ22๎‚ถ=1๐‘ž๎€œ๐‘‡0||||ฬ‡๐‘ข1ฬ‡๐‘ฃ(๐‘ก)+1(๐‘ก)2||||๐‘ž1๐‘‘๐‘ก+๐‘๎€œ๐‘‡0||||ฬ‡๐‘ข2ฬ‡๐‘ฃ(๐‘ก)+2(๐‘ก)2||||๐‘โ‰ค2๐‘‘๐‘ก๐‘žโˆ’1๐‘ž๎€œ๐‘‡012๐‘ž||ฬ‡๐‘ข1||(๐‘ก)๐‘ž2๐‘‘๐‘ก+๐‘žโˆ’1๐‘ž๎€œ๐‘‡012๐‘ž||ฬ‡๐‘ฃ1||(๐‘ก)๐‘ž+2๐‘‘๐‘ก๐‘โˆ’1๐‘๎€œ๐‘‡012๐‘||ฬ‡๐‘ข2||(๐‘ก)๐‘2๐‘‘๐‘ก+๐‘โˆ’1๐‘๎€œ๐‘‡012๐‘||ฬ‡๐‘ฃ2||(๐‘ก)๐‘โ‰ค1๐‘‘๐‘ก๎€œ2๐‘ž๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)๐‘ž1๐‘‘๐‘ก+๎€œ2๐‘ž๐‘‡0||ฬ‡๐‘ฃ1||(๐‘ก)๐‘ž+1๐‘‘๐‘ก๎€œ2๐‘๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘1๐‘‘๐‘ก+๎€œ2๐‘๐‘‡0||ฬ‡๐‘ฃ2||(๐‘ก)๐‘=๐œ™๐‘‘๐‘ก1๎€ท๐‘ข1,๐‘ข2๎€ธ+๐œ™1๎€ท๐‘ฃ1,๐‘ฃ2๎€ธ2,(3.2) then ๐œ™1 is convex. Moreover, by [33], we know that ๐œ™1 is continuous, and so, it is lower semicontinuous. Thus, it follows from [3, Theoremโ€‰โ€‰1.2] that ๐œ™1 is weakly lower continuous. By assumption (A), it is easy to verify that ๐œ™2(๐‘ข1,๐‘ข2) is weakly continuous. We omit the details. Let ๐œ“1๎€ท๐‘ข1๎€ธ=๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ,๐œ“2๎€ท๐‘ข2๎€ธ=๐‘˜๎“๐‘š=1๐พ๐‘š๎€ท๐‘ข2๎€ท๐‘ ๐‘š.๎€ธ๎€ธ(3.3) Next, we show that ๐œ“1 and ๐œ“2 are weakly continuous on ๐‘Š๐‘‡1,๐‘ž and ๐‘Š๐‘‡1,๐‘, respectively. In fact, if ๐‘ข1๐‘›โ‡€๐‘ข1weaklyin๐‘Š๐‘‡1,๐‘,as๐‘›โŸถโˆž,(3.4) then by in [3, Propositionโ€‰โ€‰1.2], we know that ๐‘ข1๐‘›โŸถ๐‘ข1stronglyin๐ถ๎€ท0,๐‘‡;โ„๐‘๎€ธ,as๐‘›โŸถโˆž.(3.5) So, there exists ๐‘€1>0 such that โ€–๐‘ข1โ€–โˆžโ‰ค๐‘€1 and โ€–๐‘ข1๐‘›โ€–โˆžโ‰ค๐‘€1, for all ๐‘›โˆˆโ„•. Thus, we have ||๐œ“1๎€ท๐‘ข1๐‘›๎€ธโˆ’๐œ“1๎€ท๐‘ข1๎€ธ||=|||||๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๐‘›๎€ท๐‘ก๐‘—โˆ’๎€ธ๎€ธ๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—|||||โ‰ค๎€ธ๎€ธ๐‘™๎“๐‘—=1||๐ผ๐‘—๎€ท๐‘ข1๐‘›๎€ท๐‘ก๐‘—๎€ธ๎€ธโˆ’๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—||=๎€ธ๎€ธ๐‘™๎“๐‘—=1||||๎€œ10๎€ทโˆ‡๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ท๐‘ข+๐‘ 1๐‘›๎€ท๐‘ก๐‘—๎€ธโˆ’๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ๎€ธ,๐‘ข1๐‘›๎€ท๐‘ก๐‘—๎€ธโˆ’๐‘ข1๎€ท๐‘ก๐‘—||||โ‰คโ€–โ€–๐‘ข๎€ธ๎€ธ๐‘‘๐‘ 1๐‘›โˆ’๐‘ข1โ€–โ€–โˆž๐‘™๎“๐‘—=1max๎€บ๐‘กโˆˆ0,3๐‘€1๎€ป๐‘Ž1(๐‘ก)โŸถ0.(3.6) Hence, ๐œ“1 is weakly continuous on ๐‘Š๐‘‡1,๐‘ž. Similarly, we can prove that ๐œ“2 is also weakly continuous on ๐‘Š๐‘‡1,๐‘. Thus, we complete the proof.

Proof of Theorem 1.1. It follows from (F1) and (2.7) that ๎€œ๐‘‡0๎€บ๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ๎€ท๐‘ข(๐‘ก)โˆ’๐น1(๐‘ก),๐‘ข2=๎€œ๎€ธ๎€ป๐‘‡0๎€œ101๐‘ ๎€ทโˆ‡๐น๐‘ฅ2๎€ท๐‘ข1(๐‘ก),๐‘ข2+๐‘ ฬƒ๐‘ข2๎€ธ(๐‘ก),๐‘ ฬƒ๐‘ข2๎€ธ=๎€œ(๐‘ก)๐‘‘๐‘ ๐‘‘๐‘ก๐‘‡0๎€œ101๐‘ ๎€ทโˆ‡๐น๐‘ฅ2๎€ท๐‘ข1(๐‘ก),๐‘ข2+๐‘ ฬƒ๐‘ข2๎€ธ(๐‘ก)โˆ’โˆ‡๐น๐‘ฅ2๎€ท๐‘ข1,๐‘ข2๎€ธ,๐‘ ฬƒ๐‘ข2๎€ธ๐‘Ÿ(๐‘ก)๐‘‘๐‘ ๐‘‘๐‘กโ‰ฅโˆ’2๐‘๎€œ๐‘‡0||ฬƒ๐‘ข2||(๐‘ก)๐‘๐‘Ÿ๐‘‘๐‘กโ‰ฅโˆ’2๐‘‡๐‘ฮ˜๎€ท๐‘,๐‘๎…ž๎€ธ๐‘(๐‘๎…ž)+1๐‘/๐‘โ€ฒ๎€œ๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘๎€ท๐‘ข๐‘‘๐‘ก,โˆ€1,๐‘ข2๎€ธโˆˆ๐‘Š,(3.7)๎€œ๐‘‡0๎€บ๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ๎€ทโˆ’๐น๐‘ข1,๐‘ข2=๎€œ๎€ธ๎€ป๐‘‘๐‘ก๐‘‡0๎€œ101๐‘ ๎€ทโˆ‡๐‘ฅ1๐น๎€ท๐‘ข1+๐‘ ฬƒ๐‘ข1(๐‘ก),๐‘ข2๎€ธ,๐‘ ฬƒ๐‘ข1๎€ธ=๎€œ(๐‘ก)๐‘‘๐‘ ๐‘‘๐‘ก๐‘‡0๎€œ101๐‘ ๎€ทโˆ‡๐‘ฅ1๐น๎€ท๐‘ข1+๐‘ ฬƒ๐‘ข1(๐‘ก),๐‘ข2๎€ธโˆ’โˆ‡๐‘ฅ1๐น๎€ท๐‘ข1,๐‘ข2๎€ธ,๐‘ ฬƒ๐‘ข1๎€ธ๐‘Ÿ(๐‘ก)๐‘‘๐‘ ๐‘‘๐‘กโ‰ฅโˆ’1๐‘ž๎€œ๐‘‡0||ฬƒ๐‘ข1||(๐‘ก)๐‘ž๐‘Ÿ๐‘‘๐‘กโ‰ฅโˆ’1๐‘‡๐‘žฮ˜๎€ท๐‘ž,๐‘ž๎…ž๎€ธ๐‘ž(๐‘žโ€ฒ+1)๐‘ž/๐‘ž๎…ž๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)๐‘ž๎€ท๐‘ข๐‘‘๐‘ก,โˆ€1,๐‘ข2๎€ธโˆˆ๐‘Š.(3.8) Hence, by (I1), (3.7), and (3.8), we have ๐œ‘๎€ท๐‘ข1,๐‘ข2๎€ธ=1๐‘ž๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)๐‘ž1๐‘‘๐‘ก+๐‘๎€œ๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘๎€œ๐‘‘๐‘ก+๐‘‡0๎€บ๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ๎€ท๐‘ข(๐‘ก)โˆ’๐น1(๐‘ก),๐‘ข2+๎€œ๎€ธ๎€ป๐‘‘๐‘ก๐‘‡0๎€บ๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ๎€ทโˆ’๐น๐‘ข1,๐‘ข2๎€ท๎€ธ๎€ป๐‘‘๐‘ก+๐‘‡๐น๐‘ข1,๐‘ข2๎€ธ+๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—+๎€ธ๎€ธ๐‘˜๎“๐‘š=1๐พ๐‘š๎€ท๐‘ข2๎€ท๐‘ ๐‘šโ‰ฅ๎ƒฉ1๎€ธ๎€ธ๐‘โˆ’๐‘Ÿ2๐‘‡๐‘ฮ˜๎€ท๐‘,๐‘๎…ž๎€ธ๐‘(๐‘โ€ฒ+1)๐‘/๐‘โ€ฒ๎ƒช๎€œ๐‘‡0||ฬ‡๐‘ข2(||๐‘ก)๐‘๎ƒฉ1๐‘‘๐‘ก+๐‘žโˆ’๐‘Ÿ1๐‘‡๐‘žฮ˜๎€ท๐‘ž,๐‘ž๎…ž๎€ธ๐‘ž(๐‘žโ€ฒ+1)๐‘ž/๐‘žโ€ฒ๎ƒช๎€œ๐‘‡0||ฬ‡๐‘ข1(||๐‘ก)๐‘ž๎€ท๐‘‘๐‘ก+๐‘‡๐น๐‘ข1,๐‘ข2๎€ธ||๐›ฝ||.โˆ’(๐‘™+๐‘˜)(3.9) Note that for ๐‘ขโˆˆ๐‘Š๐‘‡1,๐‘, โ€–๐‘ขโ€–๐‘Š๐‘‡1,๐‘๎‚ต||โŸถโˆžโŸบ๐‘ข||๐‘+๎€œ๐‘‡0||||ฬ‡๐‘ข(๐‘ก)๐‘๎‚ถ๐‘‘๐‘ก1/๐‘,โŸถโˆž(3.10) and for ๐‘ฃโˆˆ๐‘Š๐‘‡1,๐‘ž, โ€–๐‘ฃโ€–๐‘Š๐‘‡1,๐‘ž๎‚ต|โŸถโˆžโŸบ๐‘ฃ|๐‘ž+๎€œ๐‘‡0||ฬ‡||๐‘ฃ(๐‘ก)๐‘ž๎‚ถ๐‘‘๐‘ก1/๐‘žโŸถโˆž.(3.11) So, (F2) and (3.9) imply that ๐œ‘๎€ท๐‘ข1,๐‘ข2๎€ธโŸถ+โˆž,asโ€–โ€–๎€ท๐‘ข1,๐‘ข2๎€ธโ€–โ€–๐‘ŠโŸถโˆž.(3.12) Thus, by Lemma 2.3, we know that ๐œ‘ has at least one critical point which minimizes ๐œ‘ on ๐‘Š.
Furthermore, if ๐ผ๐‘—(๐‘ข1(๐‘ก๐‘—))โ‰ก0(๐‘—โˆˆ๐ต) and ๐พ๐‘š(๐‘ข2(๐‘ ๐‘š))โ‰ก0(๐‘šโˆˆ๐ถ), then system (1.1) reduces to (1.7). When (F3) also holds, we will use Lemma 2.4 to obtain more critical points of ๐œ‘. Let ๐‘‹=๐‘Š, โ€‰โ€‰๐‘‹2=โ„๐‘ร—โ„๐‘ and ๐‘‹1=๎‚‹๎‚‹๐‘Š๐‘Š=๐‘‡1,๐‘žร—๎‚‹๐‘Š๐‘‡1,๐‘.
By (3.9), we know that ๐œ‘(๐‘ข1,๐‘ข2)โ†’+โˆž as โ€–(๐‘ข1,๐‘ข2)โ€–๐‘Šโ†’โˆž. So, ๐œ‘ satisfies (PS) condition and is bounded below. Take ๐œŒ=๐›ฟ/๐‘1, where ๐‘1 is a positive constant such that โ€–๐‘ข1โ€–โˆžโ‰ค๐‘1โ€–๐‘ข1โ€–๐‘Š๐‘‡1,๐‘žโ‰ค๐‘1โ€–๐‘ขโ€–๐‘Š and โ€–๐‘ข2โ€–โˆžโ‰ค๐‘1โ€–๐‘ข2โ€–๐‘Š๐‘‡1,๐‘โ‰ค๐‘1โ€–๐‘ขโ€–๐‘Š for all (๐‘ข1,๐‘ข2)โˆˆ๐‘Š. It follows from (F3) and Lemma 2.1 that ๐œ‘๎€ท๐‘ข1,๐‘ข2๎€ธ=1q๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)๐‘ž1๐‘‘๐‘ก+๐‘๎€œ๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘๎€œ๐‘‘๐‘ก+๐‘‡0๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธโ‰ฅ1(๐‘ก)๐‘‘๐‘ก๐‘ž๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)๐‘ž1๐‘‘๐‘ก+๐‘๎€œ๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘๎€œ๐‘‘๐‘กโˆ’๐‘Ž๐‘‡0||๐‘ข1||(๐‘ก)๐‘ž๎€œ๐‘‘๐‘กโˆ’๐‘๐‘‡0||๐‘ข2||(๐‘ก)๐‘โ‰ฅ1๐‘‘๐‘ก๐‘ž๎€œ๐‘‡0||ฬ‡๐‘ข1(||๐‘ก)๐‘ž1๐‘‘๐‘ก+๐‘๎€œ๐‘‡0||ฬ‡๐‘ข2(||๐‘ก)๐‘๐‘‡๐‘‘๐‘กโˆ’๐‘Ž๐‘žฮ˜๎€ท๐‘ž,๐‘ž๎…ž๎€ธ(๐‘žโ€ฒ+1)๐‘ž/๐‘ž๎…ž๎€œ๐‘‡0||ฬ‡๐‘ข1(||๐‘ก)๐‘ž๐‘‡๐‘‘๐‘กโˆ’๐‘๐‘ฮ˜๎€ท๐‘,๐‘๎…ž๎€ธ(๐‘โ€ฒ+1)๐‘/๐‘๎…ž๎€œ๐‘‡0||ฬ‡๐‘ข2||(๐‘ก)๐‘๎€ท๐‘ข๐‘‘๐‘ก,โˆ€1,๐‘ข2๎€ธโˆˆ๐‘‹1.(3.13) Since ๐‘Žโ‰ค(๐‘žโ€ฒ+1)๐‘ž/๐‘žโ€ฒ/(๐‘ž๐‘‡๐‘žฮ˜(๐‘ž,๐‘ž๎…ž)) and ๐‘โ‰ค(๐‘โ€ฒ+1)๐‘/๐‘โ€ฒ/(๐‘๐‘‡๐‘ฮ˜(๐‘,๐‘๎…ž)), (3.13) implies that ๐œ‘(๐‘ข1,๐‘ข2)โ‰ฅ0 for all (๐‘ข1,๐‘ข2)โˆˆ๐‘‹1 with โ€–๐‘ขโ€–๐‘Šโ‰ค๐œŒ. By (F3), it is easy to obtain that ๐œ‘(๐‘ข1,๐‘ข2)โ‰ค0, for all (๐‘ข1,๐‘ข2)โˆˆ๐‘‹2 with โ€–๐‘ขโ€–๐‘Šโ‰ค๐œŒ.
If inf{๐œ‘(๐‘ข1,๐‘ข2)โˆถ(๐‘ข1,๐‘ข2)โˆˆ๐‘Š}=0, then from above, we have ๐œ‘(๐‘ข1,๐‘ข2)=0 for all (๐‘ข1,๐‘ข2)โˆˆ๐‘‹2 with โ€–(๐‘ข1,๐‘ข2)โ€–๐‘Šโ‰ค๐œŒ. Hence, all (๐‘ข1,๐‘ข2)โˆˆ๐‘‹2 with โ€–(๐‘ข1,๐‘ข2)โ€–๐‘Šโ‰ค๐œŒ are minimal points of ๐œ‘, which implies that ๐œ‘ has infinitely many critical points. If inf{๐œ‘(๐‘ข1,๐‘ข2)โˆถ(๐‘ข1,๐‘ข2)โˆˆ๐‘Š}<0, then by Lemma 2.4, ๐œ‘ has at least two nonzero critical points. Hence, system (1.7) has at least two nontrivial solutions in ๐‘Š. We complete our proof.

Proof of Theorem 1.3. We only need to use (1.18) and (1.19) to replace (2.6) and (2.7) in the proof Theorem 1.1 with ๐‘=๐‘ž=2, ๐น(๐‘ก,๐‘ข1,๐‘ข2)=๐น1(๐‘ข1) and ๐พ๐‘š(๐‘ข2)โ‰ก0(๐‘šโˆˆ๐ถ). It is easy. So, we omit it.

Lemma 3.2. Under the assumptions of Theorem 1.4, the functional ๐œ‘1 defined by ๐œ‘1๎€ท๐‘ข1๎€ธ=12๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๎€œ๐‘‘๐‘ก+๐‘‡0๐น1๎€ท๐‘ข1๎€ธ(๐‘ก)๐‘‘๐‘ก+๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ๐‘‘๐‘ก(3.14) satisfies (PS) condition.

Proof. Suppose that {๐‘ข1๐‘›} is a (PS) sequence for ๐œ‘1; that is, there exists ๐ท1>0 such that ||๐œ‘๎€ท๐‘ข1๐‘›๎€ธ||โ‰ค๐ท1,โˆ€๐‘›โˆˆโ„•,๐œ‘๎…ž๎€ท๐‘ข1๐‘›๎€ธโŸถ0,as๐‘›โŸถโˆž.(3.15) Hence, for ๐‘› large enough, we have โ€–๐œ‘๎…ž(๐‘ข1๐‘›)โ€–โ‰ค1. It follows from (F1)๎…ž๎…ž, (I2), and (1.18) that โ€–โ€–ฬƒ๐‘ข1๐‘›โ€–โ€–๐‘Š๐‘‡1,2โ‰ฅ๎ซ๐œ‘๎…ž1๎€ท๐‘ข1๐‘›๎€ธ,ฬƒ๐‘ข1๐‘›๎ฌ=๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๎€œ๐‘‘๐‘ก+๐‘‡0๎€ทโˆ‡๐‘ฅ1๐น1๎€ท๐‘ข1๐‘›๎€ธ(๐‘ก),ฬƒ๐‘ข1๐‘›๎€ธ+(๐‘ก)๐‘‘๐‘ก๐‘™๎“๐‘—=1๎€ทโˆ‡๐ผ๐‘—๎€ท๐‘ข1๐‘›๎€ท๐‘ก๐‘—๎€ธ๎€ธ,ฬƒ๐‘ข1๐‘›๎€ท๐‘ก๐‘—=๎€œ๎€ธ๎€ธ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๎€œ๐‘‘๐‘ก+๐‘‡0๎€ทโˆ‡๐‘ฅ1๐น1๎€ท๐‘ข1๐‘›๎€ธ(๐‘ก)โˆ’โˆ‡๐‘ฅ1๐น1๎€ท๐‘ข1๐‘›๎€ธ(๐‘ก),ฬƒ๐‘ข1๐‘›๎€ธ+(๐‘ก)๐‘‘๐‘ก๐‘™๎“๐‘—=1๎€ทโˆ‡๐ผ๐‘—๎€ท๐‘ข1๐‘›๎€ท๐‘ก๐‘—๎€ธ๎€ธ,ฬƒ๐‘ข1๐‘›๎€ท๐‘ก๐‘—โ‰ฅ๎€œ๎€ธ๎€ธ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๐‘‡๐‘‘๐‘กโˆ’๐‘Ÿ24๐œ‹2๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2โ€–โ€–๐‘‘๐‘กโˆ’ฬƒ๐‘ข1๐‘›โ€–โ€–โˆž๐‘™๎“๐‘—=1๐‘‘๐‘—โ‰ฅ๎‚ธ๐‘‡1โˆ’๐‘Ÿ24๐œ‹2๎‚น๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›(||๐‘ก)2๎‚€๐‘‡๐‘‘๐‘กโˆ’๎‚121/2๎‚ต๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›(||๐‘ก)2๎‚ถ๐‘‘๐‘ก๐‘™1/2๎“๐‘—=1๐‘‘๐‘—,(3.16) for ๐‘› large enough. By (1.18), we have โ€–โ€–ฬƒ๐‘ข1๐‘›โ€–โ€–๐‘Š๐‘‡1,2โ‰ค๎‚ธ๐‘‡24๐œ‹2๎‚น+11/2๎‚ต๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๎‚ถ๐‘‘๐‘ก1/2,(3.17) and (3.16), (3.17), and ๐‘Ÿ<4๐œ‹2/๐‘‡2 imply that there exists ๐ท2,๐ท3>0 such that ๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๐‘‘๐‘กโ‰ค๐ท2,โ€–โ€–ฬƒ๐‘ข1๐‘›โ€–โ€–๐‘Š๐‘‡1,2โ‰ค๐ท3.(3.18) It follows from (F4), (3.15), (I3), (1.18), and (3.18) that โˆ’D1โ‰ค๐œ‘1๎€ท๐‘ข1๐‘›๎€ธ=12๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๎€œ๐‘‘๐‘ก+๐‘‡0๐น1๎€ท๐‘ข1๐‘›๎€ธ(๐‘ก)๐‘‘๐‘ก+๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—โ‰ค1๎€ธ๎€ธ2๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๐‘‘t+1๐œ‡๎€œ๐‘‡0๐น1๎€ท๐œ†๐‘ข1๐‘›๎€ธ๎€œ๐‘‘๐‘กโˆ’๐‘‡0๐น1๎€ทโˆ’ฬƒ๐‘ข1๐‘›๎€ธ(๐‘ก)๐‘‘๐‘ก+๐‘™๎“๐‘—=1๐›พ๐‘—=12๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๐‘‡๐‘‘๐‘ก+๐œ‡๐น1๎€ท๐œ†๐‘ข1๐‘›๎€ธโˆ’๐‘‡๐น1๎€œ(0)โˆ’๐‘‡0๎€บ๐น1๎€ทโˆ’ฬƒ๐‘ข1๐‘›๎€ธ(๐‘ก)โˆ’๐น1๎€ป(0)๐‘‘๐‘ก+๐‘™๎“๐‘—=1๐›พ๐‘—=12๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๐‘‡๐‘‘๐‘ก+๐œ‡๐น1๎€ท๐œ†๐‘ข1๐‘›๎€ธโˆ’๐‘‡๐น1(0)+๐‘™๎“๐‘—=1๐›พ๐‘—โˆ’๎€œ๐‘‡0๎€œ101๐‘ ๎€ทโˆ‡๐น1๎€ทโˆ’๐‘ ฬƒ๐‘ข1๐‘›๎€ธ(๐‘ก)โˆ’โˆ‡๐น1(0),โˆ’๐‘ ฬƒ๐‘ข1๐‘›๎€ธโ‰ค1(๐‘ก)๐‘‘๐‘ ๐‘‘๐‘ก2๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๐‘‡๐‘‘๐‘ก+๐œ‡๐น1๎€ท๐œ†๐‘ข1๐‘›๎€ธ๎€œ+๐‘Ÿ๐‘‡0๎€œ10๐‘ ||ฬƒ๐‘ข1๐‘›||(๐‘ก)2๐‘‘๐‘ ๐‘‘๐‘กโˆ’๐‘‡๐น1(0)+๐‘™๎“๐‘—=1๐›พ๐‘—โ‰ค12๎€œ๐‘‡0||ฬ‡๐‘ข1๐‘›||(๐‘ก)2๐‘‡๐‘‘๐‘ก+๐œ‡๐น1๎€ท๐œ†๐‘ข1๐‘›๎€ธ+๐‘Ÿ2๎€œ๐‘‡0||ฬƒ๐‘ข1๐‘›||(๐‘ก)2๐‘‘๐‘กโˆ’๐‘‡๐น1(0)+๐‘™๎“๐‘—=1๐›พ๐‘—โ‰คmax{1,๐‘Ÿ}2โ€–โ€–ฬƒ๐‘ข1๐‘›โ€–โ€–2๐‘Š๐‘‡1,2+๐‘‡๐œ‡๐น1๎€ท๐œ†๐‘ข1๐‘›๎€ธโˆ’๐‘‡๐น1(0)+๐‘™๎“๐‘—=1๐›พ๐‘—โ‰คmax{1,๐‘Ÿ}2๐ท๐‘ž3+๐‘‡๐œ‡๐น1๎€ท๐œ†๐‘ข1๐‘›๎€ธโˆ’๐‘‡๐น1(0)+๐‘™๎“๐‘—=1๐›พ๐‘—,(3.19) for all ๐‘› and (3.19) and (F5) imply that {๐‘ข1๐‘›} is bounded. Combining (3.18), we know that {๐‘ข1๐‘›} is a bounded sequence. Similar to the argument in [25], it is easy to obtain that ๐œ‘ satisfies (PS) condition.
Proof of Theorem 1.4. From (I3) and (F5), it is easy to see that for ๐‘ฅ1โˆˆโ„๐‘, ๐œ‘1๎€ท๐‘ฅ1๎€ธโŸถโˆ’โˆž,as||๐‘ฅ1||โŸถโˆž.(3.20) For all ๐‘ข1โˆˆ๎‚‹๐‘Š๐‘‡1,2, by (1.18), (F1)๎…ž๎…ž and (I3), we have ๐œ‘1๎€ท๐‘ข1๎€ธ=12๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๎€œ๐‘‘๐‘ก+๐‘‡0๐น1๎€ท๐‘ข1๎€ธ(๐‘ก)๐‘‘๐‘ก+๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—=1๎€ธ๎€ธ2๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๎€œ๐‘‘๐‘ก+T0๎€บ๐น1๎€ท๐‘ข1๎€ธ(๐‘ก)โˆ’๐น1๎€ป(0)๐‘‘๐‘ก+๐‘‡๐น1(0)+๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—=1๎€ธ๎€ธ2๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๎€œ๐‘‘๐‘ก+๐‘‡0๎€œ10๎€ทโˆ‡๐น1๐‘ฅ1๎€ท๐‘ ๐‘ข1๎€ธ(๐‘ก),๐‘ข1๎€ธ(๐‘ก)๐‘‘๐‘ ๐‘‘๐‘ก+๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ+๐‘‡๐น1=1(0)2๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๎€œ๐‘‘๐‘ก+๐‘‡0๎€œ101๐‘ ๎€ทโˆ‡๐น1๐‘ฅ1๎€ท๐‘ ๐‘ข1๎€ธ(๐‘ก)โˆ’โˆ‡๐น1๐‘ฅ1(0),๐‘ ๐‘ข1๎€ธ+(๐‘ก)๐‘‘๐‘ ๐‘‘๐‘ก๐‘™๎“๐‘—=1๐ผ๐‘—๎€ท๐‘ข1๎€ท๐‘ก๐‘—๎€ธ๎€ธ+๐‘‡๐น1โ‰ฅ1(0)2๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๐‘Ÿ๐‘‘๐‘กโˆ’12๎€œ๐‘‡0||๐‘ข1||(๐‘ก)2๐‘‘๐‘ก+๐‘‡๐น1(0)โˆ’๐‘™๎“๐‘—=1๐‘๐‘—||๐‘ข1๎€ท๐‘ก๐‘—๎€ธ||๐›ผ๐‘—โˆ’๐‘™๎“๐‘—=1๐‘๐‘—โ‰ฅ12๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๐‘Ÿ๐‘‘๐‘กโˆ’1๐‘‡24๐œ‹2๎€œ๐‘‡0||ฬ‡๐‘ข1(t)||2๐‘‘๐‘ก+๐‘‡๐น1(0)โˆ’๐‘™๎“๐‘—=1๐‘๐‘—โ€–โ€–๐‘ข1โ€–โ€–๐›ผ๐‘—โˆžโˆ’๐‘™๎“๐‘—=1๐‘๐‘—โ‰ฅ๎‚ต12โˆ’๐‘Ÿ1๐‘‡24๐œ‹2๎‚ถ๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๐‘‘๐‘ก+๐‘‡๐น1โˆ’๎‚€๐‘‡(0)๎‚12๐›ผ๐‘—๐‘™/2๎“๐‘—=1๐‘๐‘—๎‚ต๎€œ๐‘‡0||ฬ‡๐‘ข1||(๐‘ก)2๎‚ถ๐‘‘๐‘ก๐›ผ๐‘—/2โˆ’๐‘™๎“๐‘—=1๐‘๐‘—.(3.21) Note that for all ๐‘ข1โˆˆ๎‚‹๐‘Š๐‘‡1,2, โ€–๐‘ข1โ€–๐‘Š๐‘‡1,2 is equivalent to โ€–ฬ‡๐‘ข1โ€–๐ฟ2. Then, ๐‘Ÿ1<4๐œ‹2/๐‘‡2, ๐›ผ๐‘—<2(๐‘—โˆˆ๐ต) and (3.21) imply that ๐œ‘1๎€ท๐‘ข1๎€ธโŸถ+โˆž,asโ€–โ€–๐‘ข1โ€–โ€–๐‘Š๐‘‡1,2โŸถโˆž,๐‘ข1โˆˆ๎‚‹๐‘Š๐‘‡1,2.(3.22) It follows from (3.20) and (3.22) that ๐œ‘1 satisfies (๐œ‘1) and (๐œ‘2) in Lemma 2.5. Combining with Lemma 3.2, Lemma 2.5 shows that ๐œ‘1 has at least one critical point. Thus, we complete the proof.

4. Examples

Example 4.1. Let ๐‘ž=4, ๐‘=2, ๐‘‡=๐œ‹, ๐‘ก1=1, and ๐‘ 1=2. Consider the following system: ๐‘‘ฮฆ๐‘‘๐‘ก4๎€ทฬ‡๐‘ข1๎€ธ(๐‘ก)=โˆ‡๐‘ข1๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2๎€ธ,(๐‘ก)a.e[],๐‘‘.๐‘กโˆˆ0,๐œ‹ฮฆ๐‘‘๐‘ก2๎€ทฬ‡๐‘ข2(๎€ธ๐‘ก)=โˆ‡๐‘ข2๐น๎€ท๐‘ข1(๐‘ก),๐‘ข2(๎€ธ,๐‘ก)a.e[],๐‘ข.๐‘กโˆˆ0,๐œ‹1(0)โˆ’๐‘ข1(๐œ‹)=ฬ‡๐‘ข1(0)โˆ’ฬ‡๐‘ข1๐‘ข(๐œ‹)=0,2(0)โˆ’๐‘ข2(๐œ‹)=ฬ‡๐‘ข2(0)โˆ’ฬ‡๐‘ข2(๐œ‹)=0,ฮ”ฮฆ4๎€ทฬ‡๐‘ข1๎€ธ(1)=ฮฆ๐‘ž๎€ทฬ‡๐‘ข1๎€ท1+๎€ธ๎€ธโˆ’ฮฆ๐‘ž๎€ทฬ‡๐‘ข1(1โˆ’)๎€ธ=โˆ‡๐ผ1๎€ท๐‘ข1๎€ธ,(1)ฮ”ฮฆ2๎€ทฬ‡๐‘ข2๎€ธ(2)=ฮฆ๐‘๎€ทฬ‡๐‘ข2๎€ท2+๎€ธ๎€ธโˆ’ฮฆ๐‘๎€ทฬ‡๐‘ข2(2โˆ’)๎€ธ=โˆ‡๐พ1๎€ท๐‘ข2๎€ธ,(2)(4.1) where ๐น(๐‘ฅ1,๐‘ฅ2)=๐‘ฅ411+๐‘ฅ412+โ‹ฏ+๐‘ฅ41๐‘+(1/๐œ‹2)(๐‘ฅ421+๐‘ฅ222+โ‹ฏ+๐‘ฅ22๐‘)โˆ’(1/2๐œ‹2)|๐‘ฅ2|2, ๐‘ฅ1=(๐‘ฅ11,๐‘ฅ12,โ€ฆ,๐‘ฅ1๐‘), ๐‘ฅ2=(๐‘ฅ21,๐‘ฅ22,โ€ฆ,๐‘ฅ2๐‘), ๐ผ1(๐‘ฅ)=๐‘’|๐‘ฅ|2, ๐พ1(๐‘ฅ)=๐‘’|๐‘ฅ|2, ๐‘ฅโˆˆโ„๐‘. It is easy to verify that all conditions of Theorem 1.1 hold so that system (4.1) has at least one weak solution. Moreover, if ๐น(๐‘ฅ1,๐‘ฅ2)=(1/๐œ‹2)(๐‘ฅ421+๐‘ฅ422+โ‹ฏ+๐‘ฅ42๐‘)โˆ’1/2๐œ‹2|๐‘ฅ2|2, ๐‘ฅ2=(๐‘ฅ21,๐‘ฅ22,โ€ฆ,๐‘ฅ2๐‘), ๐ผ1(๐‘ฅ)=0 and ๐พ1(๐‘ฅ)=0, ๐‘ฅโˆˆโ„๐‘, then system (4.1) has at least two nonzero solutions.

Example 4.2. Let ๐‘‡=2, ๐‘ก1=1. Consider the following autonomous second-order Hamiltonian system with impulsive effects: ฬˆ๐‘ข(๐‘ก)=โˆ‡๐‘ข๐น(๐‘ข(๐‘ก)),a.e[],๐‘ข๎€ท1.๐‘กโˆˆ0,2(0)โˆ’๐‘ข(2)=ฬ‡๐‘ข(0)โˆ’ฬ‡๐‘ข(2)=0,ฬ‡๐‘ข+๎€ธโˆ’ฬ‡๐‘ข(1โˆ’)=โˆ‡๐ผ1(๐‘ข(1)),(4.2) where ๐น(๐‘ง)=๐‘ง41+๐‘ง22+โ‹ฏ+๐‘ง2๐‘โˆ’1/2|๐‘ง|2,๐ผ1(๐‘ง)=๐‘’|๐‘ง|2, ๐‘ง=(๐‘ง1,โ€ฆ,๐‘ง๐‘)๐œโˆˆโ„๐‘. It is easy to verify that all conditions of Theorem 1.3 hold so that system (4.2) has at least one weak solution. Moreover, if ๐น(๐‘ง)=๐‘ง41+๐‘ง42+โ‹ฏ+๐‘ง4๐‘โˆ’1/2|๐‘ง|2 and ๐ผ1(๐‘ง)=0, ๐‘งโˆˆโ„๐‘, then system (4.2) has at least two nonzero solutions.

Example 4.3. Let ๐‘‡=๐œ‹, ๐‘ก1=2. Consider the following autonomous second-order Hamiltonian system with impulsive effects: ฬˆ๐‘ข(๐‘ก)=โˆ‡๐‘ข๐น(๐‘ข(๐‘ก)),a.e[],๎€ท2.๐‘กโˆˆ0,๐œ‹๐‘ข(0)โˆ’๐‘ข(๐œ‹)=ฬ‡๐‘ข(0)โˆ’ฬ‡๐‘ข(๐œ‹)=0,ฬ‡๐‘ข+๎€ธโˆ’ฬ‡๐‘ข(2โˆ’)=โˆ‡๐ผ1(๐‘ข(2)),(4.3) where ๐น(๐‘ง)=โˆ’|๐‘ง|2,๐ผ1(๐‘ง)=2sin๐‘ง1, ๐‘ง=(๐‘ง1,โ€ฆ,๐‘ง๐‘)๐œโˆˆโ„๐‘. It is easy to verify that all conditions of Theorem 1.4 hold so that system (4.3) has at least one weak solution.

References

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