Abstract

By using the variational method, some existence theorems are obtained for periodic solutions of autonomous (𝑞,𝑝)-Laplacian system with impulsive effects.

1. Introduction

Let 𝐵={1,2,,𝑙}, 𝐶={1,2,,𝑘}, 𝑙,𝑘.

In this paper, we consider the following system: 𝑑Φ𝑑𝑡𝑞̇𝑢1(𝑡)=𝑢1𝐹𝑢1(𝑡),𝑢2,(𝑡)a.e[],𝑑.𝑡0,𝑇Φ𝑑𝑡𝑝̇𝑢2(𝑡)=𝑢2𝐹𝑢1(𝑡),𝑢2,(𝑡)a.e[],𝑢.𝑡0,𝑇1(0)𝑢1(𝑇)=̇𝑢1(0)̇𝑢1𝑢(𝑇)=0,2(0)𝑢2(𝑇)=̇𝑢2(0)̇𝑢2(𝑇)=0,ΔΦ𝑞̇𝑢1𝑡𝑗=Φ𝑞̇𝑢1𝑡+𝑗Φ𝑞̇𝑢1𝑡𝑗=𝐼𝑗𝑢1𝑡𝑗,𝑗𝐵,ΔΦ𝑝̇𝑢2𝑠𝑚=Φ𝑝̇𝑢2𝑠+𝑚Φ𝑝̇𝑢2𝑠𝑚=𝐾𝑚𝑢2𝑠𝑚,𝑚𝐶,(1.1) where 𝑝>1,𝑞>1, 𝑇>0, 𝑢(𝑡)=(𝑢1(𝑡),𝑢2(𝑡))=(𝑢11(𝑡),𝑢21(𝑡),,𝑢𝑁1(𝑡),𝑢12(𝑡),𝑢22(𝑡),,𝑢𝑁2(𝑡))𝜏,  𝑡𝑗(𝑗=1,2,,𝑙), and 𝑠𝑚(𝑚=1,2,,𝑘) are the instants where the impulses occur and 0=𝑡0<𝑡1<𝑡2<<𝑡𝑙<𝑡𝑙+1=𝑇,0=𝑠0<𝑠1<𝑠2<<𝑠𝑘<𝑠𝑘+1=𝑇, 𝐼𝑗𝑁(𝑗𝐵), and 𝐾𝑚𝑁(𝑚𝐶) are continuously differentiableΦ𝜇(𝑧)=|𝑧|𝜇2𝑧=𝑁𝑖=1𝑧2𝑖(𝜇2)/2𝑧1𝑧𝑁,𝜇,𝜇>1,𝐼𝑗(𝑥)=𝜕𝐼𝑗𝜕𝑥1𝜕𝐼𝑗𝜕𝑥𝑁,𝐾𝑚(𝑥)=𝜕𝐾𝑚𝜕𝑥1𝜕𝐾𝑚𝜕𝑥𝑁,(1.2) and 𝐹𝑁×𝑁 satisfies the following assumption.(A)𝐹(𝑥) is continuously differentiable in (𝑥1,𝑥2), and there exist 𝑎1,𝑎2𝐶(+,+) such that||𝐹𝑥1,𝑥2||𝑎1||𝑥1||+𝑎2||𝑥2||,||𝑥𝐹1,𝑥2||𝑎1||𝑥1||+𝑎2||𝑥2||,||𝐼𝑗𝑥1||𝑎1||𝑥1||,||𝐼𝑗𝑥1||𝑎1||𝑥1||||𝐾,𝑗𝐵,𝑚𝑥2||𝑎2||𝑥2||,||𝐾𝑚𝑥2||𝑎2||𝑥2||,𝑚𝐶,(1.3)

for all 𝑥=(𝑥1,𝑥2)𝑁×𝑁.

When 𝑝=𝑞=2, 𝐼𝑗0(𝑗𝐵), 𝐾𝑚0(𝑚𝐶), and 𝐹(𝑢1,𝑢2)=𝐹1(𝑢1), system (1.1) reduces to the following autonomous second-order Hamiltonian system:̈𝑢1(𝑡)=𝑢1𝐹1𝑢1,(𝑡)a.e[],𝑢.𝑡0,𝑇1(0)𝑢1(𝑇)=̇𝑢1(0)̇𝑢1(𝑇)=0.(1.4) There have been lots of results about the existence of periodic solutions for system (1.4) and nonautonomous second order Hamiltonian system̈𝑢1(𝑡)=𝑢1𝐹1𝑡,𝑢1,(𝑡)a.e[],𝑢.𝑡0,𝑇1(0)𝑢1(𝑇)=̇𝑢1(0)̇𝑢1(𝑇)=0,(1.5) (e.g., see [121]). Many solvability conditions have been given, for instance, coercive condition, subquadratic condition, superquadratic condition, convex condition, and so on.

When 𝑝=𝑞=2, 𝐼𝑗0(𝑗𝐵), 𝐾𝑚0(𝑚𝐶), and 𝐹(𝑢1,𝑢2)=𝐹1(𝑢1), system (1.1) reduces to the following autonomous second-order Hamiltonian system with impulsive effects:̈𝑢1(𝑡)=𝑢1𝐹1𝑢1,(𝑡)a.e[],𝑢.𝑡0,𝑇1(0)𝑢1(𝑇)=̇𝑢1(0)̇𝑢1(𝑇)=0,̇𝑢1𝑡+𝑗̇𝑢1𝑡𝑗=𝐼𝑗𝑢1𝑡𝑗.(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 [2228]. 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 𝐼𝑗0(𝑗𝐵) and 𝐾𝑚0(𝑚𝐶), system (1.1) reduces to the following system: 𝑑Φ𝑑𝑡𝑞̇𝑢1(𝑡)=𝑢1𝐹𝑢1(𝑡),𝑢2,(𝑡)a.e[],𝑑.𝑡0,𝑇Φ𝑑𝑡𝑝̇𝑢2(𝑡)=𝑢2𝐹𝑢1(𝑡),𝑢2,(𝑡)a.e[],𝑢.𝑡0,𝑇1(0)𝑢1(𝑇)=̇𝑢1(0)̇𝑢1𝑢(𝑇)=0,2(0)𝑢2(𝑇)=̇𝑢2(0)̇𝑢2(𝑇)=0.(1.7) In [29, 30], Paşca and Tang obtained some existence results for system (1.7) by using the least action principle and saddle point theorem. Motivated by [17, 2230], 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 𝐺𝑁 is called to be (𝜆,𝜇)-quasiconcave if 𝐺(𝜆(𝑥+𝑦))𝜇(𝐺(𝑥)+𝐺(𝑦)),(1.8) for some 𝜆,𝜇>0 and 𝑥,𝑦𝑁.

Next, we state our main results.

Theorem 1.1. Let 𝑞 and 𝑝 be such that 1/𝑞+1/𝑞=1 and 1/𝑝+1/𝑝=1. Suppose 𝐹 satisfies assumption (A) and the following conditions: (F1)there exist 0<𝑟1<𝑞+1𝑞/𝑞𝑇𝑞Θ(𝑞,𝑞),0<𝑟2<𝑝+1𝑝/𝑝𝑇𝑝Θ(𝑝,𝑝),(1.9) such that 𝑥1𝐹𝑥1,𝑥2𝑦1𝐹𝑦1,𝑦2,𝑥1𝑦1𝑟1||x1𝑦1||𝑞𝑥,1,𝑥2,𝑦1,𝑦2𝑁×𝑁,𝑥2𝐹𝑥1,𝑥2𝑦2𝐹𝑦1,𝑦2,𝑥2𝑦2𝑟2||𝑥2𝑦2||𝑝𝑥,1,𝑥2,𝑦1,𝑦2𝑁×𝑁,(1.10) where Θ𝑞,𝑞=10𝑠𝑞+1+(1𝑠)𝑞+1𝑞/𝑞Θ𝑑𝑠,𝑝,𝑝=10𝑠𝑝+1+(1𝑠)𝑝+1𝑝/𝑝𝑑𝑠,(1.11)(F2)𝐹(𝑥)+,as|𝑥|,where𝑥=(𝑥1,𝑥2), (I1) there exists 𝛽 such that 𝐼𝑗(𝑥)𝛽,𝑥𝑁𝐾,𝑗𝐵,𝑚(𝑥)𝛽,𝑥𝑁,𝑚𝐶.(1.12)Then, system (1.1) has at least one solution in 𝑊𝑇1,𝑞×𝑊𝑇1,𝑝, where 𝑊𝑇1,𝑠={𝑢[0,𝑇]𝑁𝑢 is absolutely continuous,𝑢(0)=𝑢(𝑇) anḋ𝑢𝐿𝑠(0,𝑇;𝑁)},𝑠.
Furthermore, if 𝐼𝑗0(𝑗𝐵), 𝐾𝑚0(𝑚𝐶) and the following condition holds:
(F3) there exist 𝛿>0, 𝑎[0,(𝑞+1)𝑞/𝑞/𝑞𝑇𝑞Θ(𝑞,𝑞)) and 𝑏[0,(𝑝+1)𝑝/𝑝/(𝑝𝑇𝑝Θ(p,𝑝))) such that ||𝑥𝑎1||𝑞||𝑥𝑏2||𝑝𝑥𝐹1,𝑥2||𝑥0,1||||𝑥𝛿,2||𝛿,(1.13) then system (1.7) has at least two nonzero solutions in 𝑊𝑇1,𝑞×𝑊𝑇1,𝑝.

When 𝑝=𝑞=2, 𝐹(𝑥1,𝑥2)=𝐹1(𝑥1), by Theorem 1.1, it is easy to get the following corollary.

Corollary 1.2. Suppose 𝐹1 satisfies the following conditions: (A)𝐹1(𝑧) is continuously differentiable in 𝑧 and there exists 𝑎1𝐶(+,+) such that ||𝐹1||(𝑧)𝑎1||(|𝑧|),𝐹1||(𝑧)𝑎1||𝐼(|𝑧|),𝑗||(𝑧)𝑎1||(|𝑧|),𝐼𝑗||(𝑧)𝑎1(|𝑧|),𝑗𝐵,(1.14) for all 𝑧𝑁. (F1) there exists 0<𝑟<6/𝑇2 such that 𝑧𝐹1(𝑧)𝑤𝐹1(𝑤),𝑧𝑤𝑟|𝑧𝑤|2,𝑧,𝑤𝑁,(1.15)(F2)𝐹1(𝑧)+,as|𝑧|,𝑧𝑁; (I1)there exists 𝛽 such that 𝐼𝑗(𝑧)𝛽,𝑧𝑁,𝑗𝐵.(1.16)Then, system (1.6) has at least one solution in 𝑊𝑇1,2. Furthermore, if 𝐼𝑗0(𝑗𝐵) and the following condition holds:(F3) there exist 𝛿>0 and 𝑎[0,(3/𝑇2)) such that 𝑎|𝑧|2𝐹1(𝑧)0,𝑧𝑁,|𝑧|𝛿,(1.17) then system (1.4) has at least two nonzero solutions in 𝑊𝑇1,2.

For the Sobolev space 𝑊𝑇1,2, one has the following sharp estimates (see in [3, Proposition  1.2]): 𝑇0||||𝑢(𝑡)2𝑇𝑑𝑡24𝜋2𝑇0||||̇𝑢(𝑡)2𝑑𝑡Wirtingersinequality,(1.18)𝑢2𝑇12𝑇0||||̇𝑢(𝑡)2𝑑𝑡Sobolevsinequality.(1.19) By the above two inequalities, we can obtain the following better results than by Corollary 1.2.

Theorem 1.3. Suppose 𝐹1 satisfies assumption (A)′, (F2)′, (I1)′ and (F1) there exists 0<𝑟<4𝜋2/𝑇2 such that (1.15) holds.Then, system (1.6) has at least one solution in 𝑊𝑇1,2. Furthermore, if 𝐼𝑗0(𝑗𝐵) and the following condition holds:(F3) there exist 𝛿>0 and 𝑎[0,(2𝜋2)/𝑇2) such that 𝑎|𝑧|2𝐹1(𝑧)0,𝑧𝑁,|𝑧|𝛿,(1.20) then system (1.4) has at least two nonzero solutions in 𝑊𝑇1,2.

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

Theorem 1.4. Suppose 𝐹1 satisfies assumption (A)′, (F1)′ ′ and the following conditions: (F4)𝐹1(𝑧) is (𝜆,𝜇)-quasiconcave on 𝑁,(F5)𝐹1(𝑧) as |𝑧|+, 𝑧𝑁,(I2) there exist 𝑑𝑗>0(𝑗𝐵) such that ||𝐼𝑗||(𝑧)𝑑𝑗,𝑧𝑁,𝑗𝐵,(1.21)(I3) there exist 𝑏𝑗>0,𝑐𝑗>0, 𝛾𝑗, 𝛼𝑗[0,2)(𝑗𝐵) such that 𝑏𝑗|z|𝛼𝑗𝑐𝑗𝐼𝑗(𝑧)𝛾𝑗,𝑧𝑁,𝑗𝐵.(1.22)Then, system (1.6) has at least one solution in 𝑊𝑇1,2.

Remark 1.5. In [17], Yang considered the second-order Hamiltonian system with no impulsive effects, that is, system (1.4). When 𝐼𝑗0(𝑗𝐵), our Theorems 1.3 and 1.4 still improve those results in [17]. To be precise, the restriction of 𝑟 is relaxed, and some unnecessary conditions in [17] are deleted. In [17], the restriction of 𝑟 is 0<𝑟<𝑇/12, which is not right. In fact, from his proof, it is easy to see that it should be 0<𝑟<12/𝑇2. Obviously, our restriction 0<𝑟<4𝜋2/𝑇2 is better. Moreover, in our Theorem 1.4, we delete such conditions (of in [17, Theorem  1]): 𝐹1(0)=0, and there exist positive constants 𝑀,𝑁 such that 𝐹1(𝑧)𝑀|𝑧|2𝑁,𝑧𝑁.(1.23) Finally, it is remarkable that Theorems 1.3 and 1.4 are also different from those results in [116]. We can find an example satisfying our Theorem 1.3 but not satisfying the results in [121]. For example, let 𝐹1𝜋(𝑧)=22𝑇2||𝑧1||4+||𝑧2||4||𝑧++𝑁||4𝜋24𝑇2|𝑧|2,(1.24) where 𝑧=(𝑧1,,𝑧𝑁)𝜏. We can also find an example satisfying our Theorem 1.4 but not satisfying the results in [121]. For example, let 𝐹1𝑟(𝑧)=2|𝑧|2,(1.25) where 12/𝑇2<𝑟<4𝜋2/𝑇2.

2. Variational Structure and Some Preliminaries

The norm in 𝑊𝑇1,𝑝 is defined by𝑢𝑊𝑇1,𝑝=𝑇0||||𝑢(𝑡)𝑝𝑑𝑡+𝑇0||||̇𝑢(𝑡)𝑝𝑑𝑡1/𝑝.(2.1) Set𝑢𝑝=𝑇0||||𝑢(𝑡)𝑝𝑑𝑡1/𝑝,𝑢=max𝑡[0,𝑇]||||.𝑢(𝑡)(2.2) Let𝑊𝑇1,𝑝=𝑢𝑊𝑇1,𝑝𝑇0.𝑢(𝑡)𝑑𝑡=0(2.3) Obviously, 𝑊𝑇1,𝑝 is a reflexive Banach space. It is easy to know that 𝑊𝑇1,𝑝 is a subset of 𝑊𝑇1,𝑝 and 𝑊𝑇1,𝑝=𝑁𝑊𝑇1,𝑝. In this paper, we will use the space 𝑊 defined by 𝑊=𝑊𝑇1,𝑞×𝑊𝑇1,𝑝𝑢,𝑢(𝑡)=1(𝑡),𝑢2,(𝑡)(2.4) with the norm (𝑢1,𝑢2)𝑊=𝑢1𝑊𝑇1,𝑞+𝑢2𝑊𝑇1,𝑝. It is clear that 𝑊 is a reflexive Banach space. Let 𝑊𝑊=𝑇1,𝑞×𝑊𝑇1,𝑝. Then, 𝑊𝑊=(𝑇1,𝑞×𝑊𝑇1,𝑝)(𝑁×𝑁).

Lemma 2.1 (see [31] or [32]). Each 𝑢𝑊𝑇1,𝑝 and each 𝑣𝑊𝑇1,𝑞 can be written as 𝑢(𝑡)=𝑢+̃𝑢(𝑡) and 𝑣(𝑡)=̃𝑣+𝑣(𝑡) with 1𝑢=𝑇𝑇0𝑢(𝑡)𝑑𝑡,𝑇0̃𝑢(𝑡)𝑑𝑡=0,1𝑣=𝑇𝑇0𝑣(𝑡)𝑑𝑡,𝑇0̃𝑣(𝑡)𝑑𝑡=0.(2.5) Then, ̃𝑢𝑇𝑝+11/𝑝𝑇0||||̇𝑢(𝑠)𝑝𝑑𝑠1/𝑝̃,𝑣𝑇𝑞+11/𝑞𝑇0||̇||𝑣(𝑠)𝑞𝑑𝑠1/𝑞,(2.6)𝑇0||||̃𝑢(𝑠)𝑝𝑇𝑑𝑠𝑝Θ𝑝,𝑝(𝑝+1)𝑝/𝑝𝑇0||||̇𝑢(𝑠)𝑝𝑑𝑠,𝑇0||̃||𝑣(𝑠)𝑞𝑇𝑑𝑠𝑞Θ𝑞,𝑞(𝑞+1)𝑞/𝑞𝑇0||̇||𝑣(𝑠)𝑞𝑑𝑠,(2.7) where Θ𝑝,𝑝=10𝑠𝑝+1+(1𝑠)𝑝+1𝑝/𝑝𝑑𝑠,Θ𝑞,𝑞=10𝑠𝑞+1+(1𝑠)𝑞+1𝑞/𝑞𝑑𝑠.(2.8)

Note that if 𝑢𝑊𝑇1,𝑝, then 𝑢 is absolutely continuous. However, we cannot guarantee that ̇𝑢 is also continuous. Hence, it is possible that ΔΦ𝑝(̇𝑢(𝑡))=Φ𝑝(̇𝑢(𝑡+))Φ𝑝(̇𝑢(𝑡))0, which results in impulsive effects.

Following the idea in [22], one takes 𝑣1𝑊𝑇1,𝑞 and multiplies the two sides of𝑑||𝑑𝑡̇𝑢1||(𝑡)𝑞2̇𝑢1(𝑡)𝑥1𝐹𝑢1(𝑡),𝑢2(𝑡)=0,(2.9)

by 𝑣1 and integrate from 0 to 𝑇, one obtains𝑇0𝑑||𝑑𝑡̇𝑢1||(𝑡)𝑞2̇𝑢1(𝑡)𝑥1𝐹𝑢1(𝑡),𝑢2𝑣(𝑡)1.(𝑡)𝑑𝑡=0(2.10) Note that 𝑣1(𝑡) is continuous. So, 𝑣1(𝑡𝑗)=𝑣1(𝑡+𝑗)=𝑣1(𝑡𝑗). Combining ̇𝑢1(0)̇𝑢1(𝑇)=0, one has 𝑇0𝑑Φ𝑞̇𝑢1(𝑡)𝑑𝑡,𝑣1(𝑡)𝑑𝑡=𝑙𝑗=0𝑡𝑗+1𝑡𝑗𝑑Φ𝑞̇𝑢1(𝑡)𝑑𝑡,𝑣1=(𝑡)𝑑𝑡𝑙𝑗=0Φ𝑞̇𝑢1𝑡𝑗+1,𝑣1𝑡𝑗+1Φ𝑞̇𝑢1𝑡+𝑗,𝑣1𝑡+𝑗𝑑𝑡𝑙𝑗=0𝑡𝑗+1𝑡𝑗Φ𝑞̇𝑢1,̇𝑣(𝑡)1=Φ(𝑡)𝑑𝑡𝑞̇𝑢1(𝑇),𝑣1Φ(𝑇)𝑞̇𝑢1(0),𝑣1(0)𝑙𝑗=1ΔΦ𝑞̇𝑢1𝑡𝑗,𝑣1𝑡𝑗𝑇0Φ𝑞̇𝑢1(,̇𝑣𝑡)1(𝑡)𝑑𝑡=𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗,𝑣1𝑡𝑗𝑇0Φ𝑞̇𝑢1,̇𝑣(𝑡)1(𝑡)𝑑𝑡.(2.11) Combining with (2.10), one has𝑇0Φ𝑞̇𝑢1,̇𝑣(𝑡)1(𝑡)𝑑𝑡+𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗,𝑣1𝑡𝑗+𝑇0𝑥1𝐹𝑢1(𝑡),𝑢2(𝑡),𝑣1(𝑡)𝑑𝑡=0.(2.12) Similarly, one can get𝑇0Φ𝑝̇𝑢2,̇𝑣(𝑡)2(𝑡)𝑑𝑡+𝑘𝑚=1𝐾𝑚𝑢2𝑠𝑚,𝑣2𝑠𝑚+𝑇0𝑥2𝐹𝑢1(𝑡),𝑢2(𝑡),𝑣2(𝑡)𝑑𝑡=0,(2.13) for all 𝑣2𝑊𝑇1,𝑝. 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 𝑢=(𝑢1,𝑢2)𝑊𝑇1,𝑞×𝑊𝑇1,𝑝 is a weak solution of system (1.1) if𝑇0Φ𝑞̇𝑢1,̇𝑣(𝑡)1(𝑡)𝑑𝑡+𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗,𝑣1𝑡𝑗=𝑇0𝑥1𝐹𝑢1(𝑡),𝑢2(𝑡),𝑣1(𝑡)𝑑𝑡,𝑇0Φ𝑝̇𝑢2,̇𝑣(𝑡)2(𝑡)𝑑𝑡+𝑘𝑚=1𝐾𝑚𝑢2𝑠𝑚,𝑣2𝑠𝑚=𝑇0𝑥2𝐹𝑢1(𝑡),𝑢2(𝑡),𝑣2(𝑡)𝑑𝑡(2.14) holds for any 𝑣=(𝑣1,𝑣2)𝑊𝑇1,𝑞×𝑊𝑇1,𝑝.
Define the functional 𝜑𝑊𝑇1,𝑞×𝑊𝑇1,𝑝 by 𝜑𝑢1,𝑢2=1𝑞𝑇0||̇𝑢1(t)||𝑞1𝑑𝑡+𝑝𝑇0||̇𝑢2||(𝑡)𝑝𝑑𝑡+𝑇0𝐹𝑢1(𝑡),𝑢2+(𝑡)𝑑𝑡𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗+𝑘𝑚=1𝐾𝑚𝑢2𝑠𝑚𝑢=𝜙1,𝑢2𝑢+𝜓1,𝑢2,(2.15) where (𝑢1,𝑢2)𝑊𝑇1,𝑞×𝑊𝑇1,𝑝, 𝜙𝑢1,𝑢2=1𝑞𝑇0||̇𝑢1||(𝑡)𝑞1𝑑𝑡+𝑝𝑇0||̇𝑢2||(𝑡)𝑝𝑑𝑡+𝑇0𝐹𝑢1(𝑡),𝑢2𝜓𝑢(𝑡)𝑑𝑡,1,𝑢2=𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗+𝑘𝑚=1𝐾𝑚𝑢2𝑠𝑚.(2.16) By assumption (A) and [33], we know that 𝜙𝐶1(𝑊𝑇1,𝑞×𝑊𝑇1,𝑝,). The continuity of 𝐼𝑗(𝑗𝐵) and 𝐾𝑚(𝑚𝐶) implies that 𝜓𝐶1(𝑊𝑇1,𝑝×𝑊𝑇1,𝑝,). So, 𝜑𝐶1(𝑊𝑇1,𝑝,), and for all (𝑣1,𝑣2)𝑊𝑇1,𝑞×𝑊𝑇1,𝑝, we have 𝜑𝑢1,𝑢2,𝑣1,𝑣2=𝑇0Φ𝑞̇𝑢1,̇𝑣(𝑡)1(𝑡)𝑑𝑡+𝑇0Φ𝑝̇𝑢2,̇𝑣(𝑡)2+(𝑡)𝑑𝑡𝑇0𝑥1𝐹𝑢1(𝑡),𝑢2(𝑡),𝑣1(𝑡)𝑑𝑡+𝑇0𝑥2𝐹𝑢1(𝑡),𝑢2(𝑡),𝑣2+(𝑡)𝑑𝑡𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗,𝑣1𝑡𝑗+𝑘𝑚=1𝐾𝑚𝑢2𝑠𝑚,𝑣2𝑠𝑚.(2.17) Definition 2.2 shows that the critical points of 𝜑 correspond to the weak solutions of system (1.1).
We will use the following lemma to seek the critical point of 𝜑.

Lemma 2.3 (see [3, Theorem  1.1]). If 𝜑 is weakly lower semicontinuous on a reflexive Banach space 𝑋 and has a bounded minimizing sequence, then 𝜑 has a minimum on 𝑋.

Lemma 2.4 (see [34]). Let 𝜑 be a 𝐶1 function on 𝑋=𝑋1𝑋2 with 𝜑(0)=0, satisfying (PS) condition, and assume that for some 𝜌>0, 𝜑(𝑢)0,for𝑢𝑋1𝜑,𝑢𝜌,(𝑢)0,for𝑢𝑋2,𝑢𝜌.(2.18) Assume also that 𝜑 is bounded below and inf𝑋𝜑<0, then 𝜑 has at least two nonzero critical points.

Lemma 2.5 (see [35, Theorem  4.6]). Let 𝑋=𝑋1𝑋2, where 𝑋 is a real Banach space and 𝑋1{0} and is finite dimensional. Suppose that 𝜑𝐶1(𝑋,) satisfies (PS)-condition and (𝜑1)there is a constant 𝛼 and a bounded neighborhood 𝐷 of 0 in 𝑋1 such that 𝜑𝜕𝐷𝛼,(𝜑2) there is a constant 𝛽>𝛼 such that 𝜑𝑋2𝛽.
Then, 𝜑 possesses a critical value 𝑐𝛽. Moreover, 𝑐 can be characterized as 𝑐=infΓmax𝑢𝐷𝜑((𝑢)),(2.19) where, Γ=𝐶.𝐷,𝑋=𝑖𝑑on𝜕𝐷(2.20)

3. Proof of Theorems

Lemma 3.1. Under assumption (A), 𝜑 is weakly lower semicontinuous on 𝑊𝑇1,𝑞×𝑊𝑇1,𝑝.

Proof. Let 𝜙1𝑢1,𝑢2=1𝑞𝑇0||̇𝑢1||(𝑡)𝑞1𝑑𝑡+𝑝𝑇0||̇𝑢2||(𝑡)𝑝𝜙𝑑𝑡,2𝑢1,𝑢2=𝑇0𝐹𝑢1(𝑡),𝑢2(𝑡)𝑑𝑡.(3.1) Since 𝜙1𝑢1+𝑣12,𝑢2+𝑣22=1𝑞𝑇0||||̇𝑢1̇𝑣(𝑡)+1(𝑡)2||||𝑞1𝑑𝑡+𝑝𝑇0||||̇𝑢2̇𝑣(𝑡)+2(𝑡)2||||𝑝2𝑑𝑡𝑞1𝑞𝑇012𝑞||̇𝑢1||(𝑡)𝑞2𝑑𝑡+𝑞1𝑞𝑇012𝑞||̇𝑣1||(𝑡)𝑞+2𝑑𝑡𝑝1𝑝𝑇012𝑝||̇𝑢2||(𝑡)𝑝2𝑑𝑡+𝑝1𝑝𝑇012𝑝||̇𝑣2||(𝑡)𝑝1𝑑𝑡2𝑞𝑇0||̇𝑢1||(𝑡)𝑞1𝑑𝑡+2𝑞𝑇0||̇𝑣1||(𝑡)𝑞+1𝑑𝑡2𝑝𝑇0||̇𝑢2||(𝑡)𝑝1𝑑𝑡+2𝑝𝑇0||̇𝑣2||(𝑡)𝑝=𝜙𝑑𝑡1𝑢1,𝑢2+𝜙1𝑣1,𝑣22,(3.2) then 𝜙1 is convex. Moreover, by [33], we know that 𝜙1 is continuous, and so, it is lower semicontinuous. Thus, it follows from [3, Theorem  1.2] that 𝜙1 is weakly lower continuous. By assumption (A), it is easy to verify that 𝜙2(𝑢1,𝑢2) is weakly continuous. We omit the details. Let 𝜓1𝑢1=𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗,𝜓2𝑢2=𝑘𝑚=1𝐾𝑚𝑢2𝑠𝑚.(3.3) Next, we show that 𝜓1 and 𝜓2 are weakly continuous on 𝑊𝑇1,𝑞 and 𝑊𝑇1,𝑝, respectively. In fact, if 𝑢1𝑛𝑢1weaklyin𝑊𝑇1,𝑝,as𝑛,(3.4) then by in [3, Proposition  1.2], we know that 𝑢1𝑛𝑢1stronglyin𝐶0,𝑇;𝑁,as𝑛.(3.5) So, there exists 𝑀1>0 such that 𝑢1𝑀1 and 𝑢1𝑛𝑀1, for all 𝑛. Thus, we have ||𝜓1𝑢1𝑛𝜓1𝑢1||=|||||𝑙𝑗=1𝐼𝑗𝑢1𝑛𝑡𝑗𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗|||||𝑙𝑗=1||𝐼𝑗𝑢1𝑛𝑡𝑗𝐼𝑗𝑢1𝑡𝑗||=𝑙𝑗=1||||10𝐼𝑗𝑢1𝑡𝑗𝑢+𝑠1𝑛𝑡𝑗𝑢1𝑡𝑗,𝑢1𝑛𝑡𝑗𝑢1𝑡𝑗||||𝑢𝑑𝑠1𝑛𝑢1𝑙𝑗=1max𝑡0,3𝑀1𝑎1(𝑡)0.(3.6) Hence, 𝜓1 is weakly continuous on 𝑊𝑇1,𝑞. Similarly, we can prove that 𝜓2 is also weakly continuous on 𝑊𝑇1,𝑝. Thus, we complete the proof.

Proof of Theorem 1.1. It follows from (F1) and (2.7) that 𝑇0𝐹𝑢1(𝑡),𝑢2𝑢(𝑡)𝐹1(𝑡),𝑢2=𝑇0101𝑠𝐹𝑥2𝑢1(𝑡),𝑢2+𝑠̃𝑢2(𝑡),𝑠̃𝑢2=(𝑡)𝑑𝑠𝑑𝑡𝑇0101𝑠𝐹𝑥2𝑢1(𝑡),𝑢2+𝑠̃𝑢2(𝑡)𝐹𝑥2𝑢1,𝑢2,𝑠̃𝑢2𝑟(𝑡)𝑑𝑠𝑑𝑡2𝑝𝑇0||̃𝑢2||(𝑡)𝑝𝑟𝑑𝑡2𝑇𝑝Θ𝑝,𝑝𝑝(𝑝)+1𝑝/𝑝𝑇0||̇𝑢2||(𝑡)𝑝𝑢𝑑𝑡,1,𝑢2𝑊,(3.7)𝑇0𝐹𝑢1(𝑡),𝑢2𝐹𝑢1,𝑢2=𝑑𝑡𝑇0101𝑠𝑥1𝐹𝑢1+𝑠̃𝑢1(𝑡),𝑢2,𝑠̃𝑢1=(𝑡)𝑑𝑠𝑑𝑡𝑇0101𝑠𝑥1𝐹𝑢1+𝑠̃𝑢1(𝑡),𝑢2𝑥1𝐹𝑢1,𝑢2,𝑠̃𝑢1𝑟(𝑡)𝑑𝑠𝑑𝑡1𝑞𝑇0||̃𝑢1||(𝑡)𝑞𝑟𝑑𝑡1𝑇𝑞Θ𝑞,𝑞𝑞(𝑞+1)𝑞/𝑞𝑇0||̇𝑢1||(𝑡)𝑞𝑢𝑑𝑡,1,𝑢2𝑊.(3.8) Hence, by (I1), (3.7), and (3.8), we have 𝜑𝑢1,𝑢2=1𝑞𝑇0||̇𝑢1||(𝑡)𝑞1𝑑𝑡+𝑝𝑇0||̇𝑢2||(𝑡)𝑝𝑑𝑡+𝑇0𝐹𝑢1(𝑡),𝑢2𝑢(𝑡)𝐹1(𝑡),𝑢2+𝑑𝑡𝑇0𝐹𝑢1(𝑡),𝑢2𝐹𝑢1,𝑢2𝑑𝑡+𝑇𝐹𝑢1,𝑢2+𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗+𝑘𝑚=1𝐾𝑚𝑢2𝑠𝑚1𝑝𝑟2𝑇𝑝Θ𝑝,𝑝𝑝(𝑝+1)𝑝/𝑝𝑇0||̇𝑢2(||𝑡)𝑝1𝑑𝑡+𝑞𝑟1𝑇𝑞Θ𝑞,𝑞𝑞(𝑞+1)𝑞/𝑞𝑇0||̇𝑢1(||𝑡)𝑞𝑑𝑡+𝑇𝐹𝑢1,𝑢2||𝛽||.(𝑙+𝑘)(3.9) Note that for 𝑢𝑊𝑇1,𝑝, 𝑢𝑊𝑇1,𝑝||𝑢||𝑝+𝑇0||||̇𝑢(𝑡)𝑝𝑑𝑡1/𝑝,(3.10) and for 𝑣𝑊𝑇1,𝑞, 𝑣𝑊𝑇1,𝑞|𝑣|𝑞+𝑇0||̇||𝑣(𝑡)𝑞𝑑𝑡1/𝑞.(3.11) So, (F2) and (3.9) imply that 𝜑𝑢1,𝑢2+,as𝑢1,𝑢2𝑊.(3.12) Thus, by Lemma 2.3, we know that 𝜑 has at least one critical point which minimizes 𝜑 on 𝑊.
Furthermore, if 𝐼𝑗(𝑢1(𝑡𝑗))0(𝑗𝐵) and 𝐾𝑚(𝑢2(𝑠𝑚))0(𝑚𝐶), then system (1.1) reduces to (1.7). When (F3) also holds, we will use Lemma 2.4 to obtain more critical points of 𝜑. Let 𝑋=𝑊,   𝑋2=𝑁×𝑁 and 𝑋1=𝑊𝑊=𝑇1,𝑞×𝑊𝑇1,𝑝.
By (3.9), we know that 𝜑(𝑢1,𝑢2)+ as (𝑢1,𝑢2)𝑊. So, 𝜑 satisfies (PS) condition and is bounded below. Take 𝜌=𝛿/𝑐1, where 𝑐1 is a positive constant such that 𝑢1𝑐1𝑢1𝑊𝑇1,𝑞𝑐1𝑢𝑊 and 𝑢2𝑐1𝑢2𝑊𝑇1,𝑝𝑐1𝑢𝑊 for all (𝑢1,𝑢2)𝑊. It follows from (F3) and Lemma 2.1 that 𝜑𝑢1,𝑢2=1q𝑇0||̇𝑢1||(𝑡)𝑞1𝑑𝑡+𝑝𝑇0||̇𝑢2||(𝑡)𝑝𝑑𝑡+𝑇0𝐹𝑢1(𝑡),𝑢21(𝑡)𝑑𝑡𝑞𝑇0||̇𝑢1||(𝑡)𝑞1𝑑𝑡+𝑝𝑇0||̇𝑢2||(𝑡)𝑝𝑑𝑡𝑎𝑇0||𝑢1||(𝑡)𝑞𝑑𝑡𝑏𝑇0||𝑢2||(𝑡)𝑝1𝑑𝑡𝑞𝑇0||̇𝑢1(||𝑡)𝑞1𝑑𝑡+𝑝𝑇0||̇𝑢2(||𝑡)𝑝𝑇𝑑𝑡𝑎𝑞Θ𝑞,𝑞(𝑞+1)𝑞/𝑞𝑇0||̇𝑢1(||𝑡)𝑞𝑇𝑑𝑡𝑏𝑝Θ𝑝,𝑝(𝑝+1)𝑝/𝑝𝑇0||̇𝑢2||(𝑡)𝑝𝑢𝑑𝑡,1,𝑢2𝑋1.(3.13) Since 𝑎(𝑞+1)𝑞/𝑞/(𝑞𝑇𝑞Θ(𝑞,𝑞)) and 𝑏(𝑝+1)𝑝/𝑝/(𝑝𝑇𝑝Θ(𝑝,𝑝)), (3.13) implies that 𝜑(𝑢1,𝑢2)0 for all (𝑢1,𝑢2)𝑋1 with 𝑢𝑊𝜌. By (F3), it is easy to obtain that 𝜑(𝑢1,𝑢2)0, for all (𝑢1,𝑢2)𝑋2 with 𝑢𝑊𝜌.
If inf{𝜑(𝑢1,𝑢2)(𝑢1,𝑢2)𝑊}=0, then from above, we have 𝜑(𝑢1,𝑢2)=0 for all (𝑢1,𝑢2)𝑋2 with (𝑢1,𝑢2)𝑊𝜌. Hence, all (𝑢1,𝑢2)𝑋2 with (𝑢1,𝑢2)𝑊𝜌 are minimal points of 𝜑, which implies that 𝜑 has infinitely many critical points. If inf{𝜑(𝑢1,𝑢2)(𝑢1,𝑢2)𝑊}<0, then by Lemma 2.4, 𝜑 has at least two nonzero critical points. Hence, system (1.7) has at least two nontrivial solutions in 𝑊. 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 𝑝=𝑞=2, 𝐹(𝑡,𝑢1,𝑢2)=𝐹1(𝑢1) and 𝐾𝑚(𝑢2)0(𝑚𝐶). It is easy. So, we omit it.

Lemma 3.2. Under the assumptions of Theorem 1.4, the functional 𝜑1 defined by 𝜑1𝑢1=12𝑇0||̇𝑢1||(𝑡)2𝑑𝑡+𝑇0𝐹1𝑢1(𝑡)𝑑𝑡+𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗𝑑𝑡(3.14) satisfies (PS) condition.

Proof. Suppose that {𝑢1𝑛} is a (PS) sequence for 𝜑1; that is, there exists 𝐷1>0 such that ||𝜑𝑢1𝑛||𝐷1,𝑛,𝜑𝑢1𝑛0,as𝑛.(3.15) Hence, for 𝑛 large enough, we have 𝜑(𝑢1𝑛)1. It follows from (F1), (I2), and (1.18) that ̃𝑢1𝑛𝑊𝑇1,2𝜑1𝑢1𝑛,̃𝑢1𝑛=𝑇0||̇𝑢1𝑛||(𝑡)2𝑑𝑡+𝑇0𝑥1𝐹1𝑢1𝑛(𝑡),̃𝑢1𝑛+(𝑡)𝑑𝑡𝑙𝑗=1𝐼𝑗𝑢1𝑛𝑡𝑗,̃𝑢1𝑛𝑡𝑗=𝑇0||̇𝑢1𝑛||(𝑡)2𝑑𝑡+𝑇0𝑥1𝐹1𝑢1𝑛(𝑡)𝑥1𝐹1𝑢1𝑛(𝑡),̃𝑢1𝑛+(𝑡)𝑑𝑡𝑙𝑗=1𝐼𝑗𝑢1𝑛𝑡𝑗,̃𝑢1𝑛𝑡𝑗𝑇0||̇𝑢1𝑛||(𝑡)2𝑇𝑑𝑡𝑟24𝜋2𝑇0||̇𝑢1𝑛||(𝑡)2𝑑𝑡̃𝑢1𝑛𝑙𝑗=1𝑑𝑗𝑇1𝑟24𝜋2𝑇0||̇𝑢1𝑛(||𝑡)2𝑇𝑑𝑡121/2𝑇0||̇𝑢1𝑛(||𝑡)2𝑑𝑡𝑙1/2𝑗=1𝑑𝑗,(3.16) for 𝑛 large enough. By (1.18), we have ̃𝑢1𝑛𝑊𝑇1,2𝑇24𝜋2+11/2𝑇0||̇𝑢1𝑛||(𝑡)2𝑑𝑡1/2,(3.17) and (3.16), (3.17), and 𝑟<4𝜋2/𝑇2 imply that there exists 𝐷2,𝐷3>0 such that 𝑇0||̇𝑢1𝑛||(𝑡)2𝑑𝑡𝐷2,̃𝑢1𝑛𝑊𝑇1,2𝐷3.(3.18) It follows from (F4), (3.15), (I3), (1.18), and (3.18) that D1𝜑1𝑢1𝑛=12𝑇0||̇𝑢1𝑛||(𝑡)2𝑑𝑡+𝑇0𝐹1𝑢1𝑛(𝑡)𝑑𝑡+𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗12𝑇0||̇𝑢1𝑛||(𝑡)2𝑑t+1𝜇𝑇0𝐹1𝜆𝑢1𝑛𝑑𝑡𝑇0𝐹1̃𝑢1𝑛(𝑡)𝑑𝑡+𝑙𝑗=1𝛾𝑗=12𝑇0||̇𝑢1𝑛||(𝑡)2𝑇𝑑𝑡+𝜇𝐹1𝜆𝑢1𝑛𝑇𝐹1(0)𝑇0𝐹1̃𝑢1𝑛(𝑡)𝐹1(0)𝑑𝑡+𝑙𝑗=1𝛾𝑗=12𝑇0||̇𝑢1𝑛||(𝑡)2𝑇𝑑𝑡+𝜇𝐹1𝜆𝑢1𝑛𝑇𝐹1(0)+𝑙𝑗=1𝛾𝑗𝑇0101𝑠𝐹1𝑠̃𝑢1𝑛(𝑡)𝐹1(0),𝑠̃𝑢1𝑛1(𝑡)𝑑𝑠𝑑𝑡2𝑇0||̇𝑢1𝑛||(𝑡)2𝑇𝑑𝑡+𝜇𝐹1𝜆𝑢1𝑛+𝑟𝑇010𝑠||̃𝑢1𝑛||(𝑡)2𝑑𝑠𝑑𝑡𝑇𝐹1(0)+𝑙𝑗=1𝛾𝑗12𝑇0||̇𝑢1𝑛||(𝑡)2𝑇𝑑𝑡+𝜇𝐹1𝜆𝑢1𝑛+𝑟2𝑇0||̃𝑢1𝑛||(𝑡)2𝑑𝑡𝑇𝐹1(0)+𝑙𝑗=1𝛾𝑗max{1,𝑟}2̃𝑢1𝑛2𝑊𝑇1,2+𝑇𝜇𝐹1𝜆𝑢1𝑛𝑇𝐹1(0)+𝑙𝑗=1𝛾𝑗max{1,𝑟}2𝐷𝑞3+𝑇𝜇𝐹1𝜆𝑢1𝑛𝑇𝐹1(0)+𝑙𝑗=1𝛾𝑗,(3.19) for all 𝑛 and (3.19) and (F5) imply that {𝑢1𝑛} is bounded. Combining (3.18), we know that {𝑢1𝑛} is a bounded sequence. Similar to the argument in [25], it is easy to obtain that 𝜑 satisfies (PS) condition.
Proof of Theorem 1.4. From (I3) and (F5), it is easy to see that for 𝑥1𝑁, 𝜑1𝑥1,as||𝑥1||.(3.20) For all 𝑢1𝑊𝑇1,2, by (1.18), (F1) and (I3), we have 𝜑1𝑢1=12𝑇0||̇𝑢1||(𝑡)2𝑑𝑡+𝑇0𝐹1𝑢1(𝑡)𝑑𝑡+𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗=12𝑇0||̇𝑢1||(𝑡)2𝑑𝑡+T0𝐹1𝑢1(𝑡)𝐹1(0)𝑑𝑡+𝑇𝐹1(0)+𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗=12𝑇0||̇𝑢1||(𝑡)2𝑑𝑡+𝑇010𝐹1𝑥1𝑠𝑢1(𝑡),𝑢1(𝑡)𝑑𝑠𝑑𝑡+𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗+𝑇𝐹1=1(0)2𝑇0||̇𝑢1||(𝑡)2𝑑𝑡+𝑇0101𝑠𝐹1𝑥1𝑠𝑢1(𝑡)𝐹1𝑥1(0),𝑠𝑢1+(𝑡)𝑑𝑠𝑑𝑡𝑙𝑗=1𝐼𝑗𝑢1𝑡𝑗+𝑇𝐹11(0)2𝑇0||̇𝑢1||(𝑡)2𝑟𝑑𝑡12𝑇0||𝑢1||(𝑡)2𝑑𝑡+𝑇𝐹1(0)𝑙𝑗=1𝑏𝑗||𝑢1𝑡𝑗||𝛼𝑗𝑙𝑗=1𝑐𝑗12𝑇0||̇𝑢1||(𝑡)2𝑟𝑑𝑡1𝑇24𝜋2𝑇0||̇𝑢1(t)||2𝑑𝑡+𝑇𝐹1(0)𝑙𝑗=1𝑏𝑗𝑢1𝛼𝑗𝑙𝑗=1𝑐𝑗12𝑟1𝑇24𝜋2𝑇0||̇𝑢1||(𝑡)2𝑑𝑡+𝑇𝐹1𝑇(0)12𝛼𝑗𝑙/2𝑗=1𝑏𝑗𝑇0||̇𝑢1||(𝑡)2𝑑𝑡𝛼𝑗/2𝑙𝑗=1𝑐𝑗.(3.21) Note that for all 𝑢1𝑊𝑇1,2, 𝑢1𝑊𝑇1,2 is equivalent to ̇𝑢1𝐿2. Then, 𝑟1<4𝜋2/𝑇2, 𝛼𝑗<2(𝑗𝐵) and (3.21) imply that 𝜑1𝑢1+,as𝑢1𝑊𝑇1,2,𝑢1𝑊𝑇1,2.(3.22) It follows from (3.20) and (3.22) that 𝜑1 satisfies (𝜑1) and (𝜑2) in Lemma 2.5. Combining with Lemma 3.2, Lemma 2.5 shows that 𝜑1 has at least one critical point. Thus, we complete the proof.

4. Examples

Example 4.1. Let 𝑞=4, 𝑝=2, 𝑇=𝜋, 𝑡1=1, and 𝑠1=2. Consider the following system: 𝑑Φ𝑑𝑡4̇𝑢1(𝑡)=𝑢1𝐹𝑢1(𝑡),𝑢2,(𝑡)a.e[],𝑑.𝑡0,𝜋Φ𝑑𝑡2̇𝑢2(𝑡)=𝑢2𝐹𝑢1(𝑡),𝑢2(,𝑡)a.e[],𝑢.𝑡0,𝜋1(0)𝑢1(𝜋)=̇𝑢1(0)̇𝑢1𝑢(𝜋)=0,2(0)𝑢2(𝜋)=̇𝑢2(0)̇𝑢2(𝜋)=0,ΔΦ4̇𝑢1(1)=Φ𝑞̇𝑢11+Φ𝑞̇𝑢1(1)=𝐼1𝑢1,(1)ΔΦ2̇𝑢2(2)=Φ𝑝̇𝑢22+Φ𝑝̇𝑢2(2)=𝐾1𝑢2,(2)(4.1) where 𝐹(𝑥1,𝑥2)=𝑥411+𝑥412++𝑥41𝑁+(1/𝜋2)(𝑥421+𝑥222++𝑥22𝑁)(1/2𝜋2)|𝑥2|2, 𝑥1=(𝑥11,𝑥12,,𝑥1𝑁), 𝑥2=(𝑥21,𝑥22,,𝑥2𝑁), 𝐼1(𝑥)=𝑒|𝑥|2, 𝐾1(𝑥)=𝑒|𝑥|2, 𝑥𝑁. 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 𝐹(𝑥1,𝑥2)=(1/𝜋2)(𝑥421+𝑥422++𝑥42𝑁)1/2𝜋2|𝑥2|2, 𝑥2=(𝑥21,𝑥22,,𝑥2𝑁), 𝐼1(𝑥)=0 and 𝐾1(𝑥)=0, 𝑥𝑁, then system (4.1) has at least two nonzero solutions.

Example 4.2. Let 𝑇=2, 𝑡1=1. Consider the following autonomous second-order Hamiltonian system with impulsive effects: ̈𝑢(𝑡)=𝑢𝐹(𝑢(𝑡)),a.e[],𝑢1.𝑡0,2(0)𝑢(2)=̇𝑢(0)̇𝑢(2)=0,̇𝑢+̇𝑢(1)=𝐼1(𝑢(1)),(4.2) where 𝐹(𝑧)=𝑧41+𝑧22++𝑧2𝑁1/2|𝑧|2,𝐼1(𝑧)=𝑒|𝑧|2, 𝑧=(𝑧1,,𝑧𝑁)𝜏𝑁. 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 𝐹(𝑧)=𝑧41+𝑧42++𝑧4𝑁1/2|𝑧|2 and 𝐼1(𝑧)=0, 𝑧𝑁, then system (4.2) has at least two nonzero solutions.

Example 4.3. Let 𝑇=𝜋, 𝑡1=2. Consider the following autonomous second-order Hamiltonian system with impulsive effects: ̈𝑢(𝑡)=𝑢𝐹(𝑢(𝑡)),a.e[],2.𝑡0,𝜋𝑢(0)𝑢(𝜋)=̇𝑢(0)̇𝑢(𝜋)=0,̇𝑢+̇𝑢(2)=𝐼1(𝑢(2)),(4.3) where 𝐹(𝑧)=|𝑧|2,𝐼1(𝑧)=2sin𝑧1, 𝑧=(𝑧1,,𝑧𝑁)𝜏𝑁. It is easy to verify that all conditions of Theorem 1.4 hold so that system (4.3) has at least one weak solution.