Infinitely many homoclinic solutions for a class of second-order Hamiltonian systems

Abstract

In this paper, we deal with the existence of infinitely many homoclinic solutions for a class of second-order Hamiltonian systems. By using the dual fountain theorem, we give some new criteria to guarantee that the second-order Hamiltonian systems have infinitely many homoclinic solutions. Some recent results are generalised and significantly improved.

MSC:34B08, 34B15, 34B37, 58E30.

1 Introduction

Consider the following second-order Hamiltonian systems:

$$\ddot{u}(t)-L(t)u(t)+W_u(t,u(t))=0,\mathrm{\forall }t\in \mathbb{R},$$

(1.1)

where $L\in C(\mathbb{R},{\mathbb{R}}^{N\times N})$ is a symmetric matrix valued function, $u\in {\mathbb{R}}^N$ and $W\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$. As usual, we say that a solution u of system (1.1) is homoclinic to zero if $u\in C^2(\mathbb{R},{\mathbb{R}}^N)$, $u\ne 0$, $u(t)\to 0$ and $\dot{u}(t)\to 0$ as $|t|\to \mathrm{\infty }$.

Inspired by the excellent monographs [1, 2], the existence of periodic solutions and homoclinic solutions for second-order Hamiltonian systems have been intensively studied in many recent papers via variational methods; see [3–30] and references therein. Recently, some researchers have begun to study the existence of solutions for second-order Hamiltonian systems with impulses by using some critical points theorems of [31, 32]; see [33, 34].

Homoclinic solutions of dynamical systems are very important in applications for a lot of reasons. They may be ‘organizing centers’ for the dynamics in their neighborhood. From their existence one may, under proper conditions, deduce the bifurcation behavior of periodic orbits or the existence of chaos nearby. In the past 20 years, with the aid of the variational methods, the existence and multiplicity of homoclinic solutions for system (1.1) have been extensively investigated by many authors; see [3, 4, 6–9, 11–30] and references therein. Most of them treated the superquadratic case [8, 15, 18, 19, 21, 24, 25, 27–29] treated subquadratic case and [9, 21, 30] treated asymptotically quadratic case. Particularly, Yang and Zhang [25] considered a superquadratic case and obtained system (1.1) has infinitely many homoclinic solutions by using the following conditions.

(A₁) $W(t,u)/{|u|}^2\to +\mathrm{\infty }$, as $|u|\to \mathrm{\infty }$ uniformly for all $t\in \mathbb{R}$, and

(A₂) $W_u(t,u(t))=o(|u|)$, as $|u|\to 0$ uniformly for all $t\in \mathbb{R}$.

Recently, Wei and Wang [22] dealt with a case that $W(t,u)=F(t,u)+G(t,u)$, where $F(t,u)$ is subquadratic and $G(t,u)$ is superquadratic, by using the following condition, they obtained system (1.1) has infinitely many homoclinic solutions.

(A₃) There exists $\nu <1$ such that

$$l(t){|t|}^{\nu -2}\to \mathrm{\infty }\text{as }|t|\to \mathrm{\infty },$$

where $l(t)={inf}_{|\xi |=1}(L(t)\xi ,\xi )$.

In [23], Yang et al. obtained the following theorems by using the variant fountain theorem.

Theorem 1.1 ([[23], Theorem 1.2])

Assume that the following conditions are satisfied:

(C₁) $L\in C(\mathbb{R},{\mathbb{R}}^{N\times N})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$ and there is a continuous function $\beta :\mathbb{R}\to \mathbb{R}$ such that $\beta (t)>0$ for all $t\in \mathbb{R}$ and $(L(t)u,u)\ge \beta (t){|u|}^2$ and $\beta (t)\to \mathrm{\infty }$ as $|t|\to \mathrm{\infty }$.

${({\mathrm{C}}_2)}^{\prime }$ $c_1^{\prime }(t){|u|}^{\gamma }\le (W_u(t,u),u)$, $|W_u(t,u)|\le c_2^{\prime }(t){|u|}^{\gamma -1}+c_3^{\prime }(t){|u|}^{\sigma -1}$ where $c_1^{\prime },c_2^{\prime },c_3^{\prime }:\mathbb{R}\to {\mathbb{R}}^{+}$ are positive continuous functions such that $c_1^{\prime },c_2^{\prime }\in L^{\frac{2}{2-\gamma }}(\mathbb{R},{\mathbb{R}}^{+})$, $c_3^{\prime }\in L^{\frac{2}{2-\sigma }}(\mathbb{R},{\mathbb{R}}^{+})$ and $1<\gamma <2$, $1<\sigma <2$ are constants, $W(t,0)=0$, $W(t,u)=W(t,-u)$ for all $(t,u)\in \mathbb{R}\times {\mathbb{R}}^N$.

Then system (1.1) possesses infinitely many homoclinic solutions.

Theorem 1.2 ([[23], Theorem 1.3])

Assume that (C₁) hold. Moreover, we assume that the following condition is satisfied:

${({\mathrm{C}}_3)}^{\prime }$ $W(t,u)=m(t){|u|}^{\gamma }+d{|u|}^q$ where $m:\mathbb{R}\to {\mathbb{R}}^{+}$ is a positive continuous function such that $m\in L^{\frac{2}{2-\gamma }}(\mathbb{R},{\mathbb{R}}^{+})$ and $1<\gamma <2$, $d\ge 0$, $q>2$ are constants.

Then system (1.1) possesses infinitely many homoclinic solutions.

Motivated by the above facts, in this paper, we will improve and generalize some results in the references that we have mentioned above.

Now, we state our main results.

Theorem 1.3 Assume that (C₁) hold. Moreover, we assume that the following conditions are satisfied:

(C₂) $W(t,u)=F(t,u)+G(t,u)$ where $F(t,0)=0$, $G(t,0)=0$ and $F,G\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$ are even in u.

(C₃) $c_1(t){|u|}^{\gamma }\le (F_u(t,u),u)$, $|F_u(t,u)|\le c_2(t){|u|}^{\gamma -1}+c_3(t){|u|}^{\sigma -1}$ where $c_1,c_2,c_3:\mathbb{R}\to {\mathbb{R}}^{+}$ are positive continuous functions such that $c_1,c_2\in L^{\frac{2}{2-\gamma }}(\mathbb{R},{\mathbb{R}}^{+})$, $c_3\in L^{\frac{2}{2-\sigma }}(\mathbb{R},{\mathbb{R}}^{+})$ and $1<\gamma <2$, $1<\sigma <2$ are constants.

(C₄) $G(t,u)\ge 0$ for all $(t,u)\in \mathbb{R}\times {\mathbb{R}}^N$ and there exists $\mu >2$ such that

$$|G_u(t,u)|\le a(t)+c(t){|u|}^{\mu -1},$$

where $a,c:\mathbb{R}\to {\mathbb{R}}^{+}$ are positive continuous functions such that $a\in L^2(\mathbb{R},{\mathbb{R}}^{+})$ and $c\in L^{\mathrm{\infty }}(\mathbb{R},{\mathbb{R}}^{+})$.

(C₅) There exist $\rho >2$ and $1<\delta <2$ such that

$$\rho G(t,u)-(G_u(t,u),u)\le h(t){|u|}^{\delta },\mathrm{\forall }(t,u)\in \mathbb{R}\times {\mathbb{R}}^N,$$

where $h:\mathbb{R}\to {\mathbb{R}}^{+}$ is a positive continuous function such that $h\in L^{\frac{2}{2-\delta }}(\mathbb{R},{\mathbb{R}}^{+})$.

Then system (1.1) possesses infinitely many homoclinic solutions.

Remark 1.1 It is clear that there are many functions satisfying (C₁) but do not satisfying (A₃); see [28].

Remark 1.2 Obviously, Theorem 1.3 generalizes Theorem 1.2 in [23], Theorem 1.1 in [29] and Theorem 1.2 in [18]. In fact, let $G(t,u)\equiv 0$, then Theorem 1.3 coincides with Theorem 1.2 in [23], and contains Theorem 1.1 in [29] and Theorem 1.2 in [18]. Furthermore, there are many functions W satisfying our Theorem 1.3 and not satisfying Theorem 1.2 in [23], Theorem 1.1 in [29] and Theorem 1.2 in [18]. For example, the function

$$W(t,u)=F(t,u)+G(t,u),$$

(1.2)

where $F(t,u)={(\frac{1}{1+t^2})}^{\frac{1}{3}}{|u|}^{\frac{4}{3}}+{(\frac{1}{1+|t|})}^{\frac{1}{3}}{|u|}^{\frac{5}{3}}$ and $G(t,u)=\frac{1}{1+t^6}{|u|}^4$.

Remark 1.3 It is easy to see that there are many functions W satisfying the conditions of Theorem 1.3 but not satisfying (A₁), (A₂), the condition ${({\mathrm{C}}_3)}^{\prime }$ in Theorem 1.2 or conditions ($R_2$) and ($R_4$) in Theorem 1.1 in [22], for example, the function (1.2) since ${(\frac{1}{1+t^2})}^{\frac{1}{3}}\to 0$, ${(\frac{1}{1+|t|})}^{\frac{1}{3}}\to 0$ and $\frac{1}{1+t^6}{|u|}^4\to 0$ as $|t|\to \mathrm{\infty }$.

Theorem 1.4 Assume that (C₁)-(C₃) hold. Moreover, we assume that the following conditions are satisfied:

(C₆) There exist $\lambda >2$ and $0\le h_1<\frac{{\beta }_0(\lambda -2)}{2}$ such that

$$\lambda G(t,u)-(G_u(t,u),u)\le h_1{|u|}^2,\mathrm{\forall }(t,u)\in \mathbb{R}\times {\mathbb{R}}^N,$$

where ${\beta }_0={min}_{t\in \mathbb{R}}\beta (t)$.

(C₇) There exists $0<\theta <{\beta }_0$ such that ${lim}_{u\to 0}\frac{G_u(t,u)}{|u|}<\theta$ uniformly for $t\in \mathbb{R}$.

(C₈) $G(t,u)\ge 0$ for all $(t,u)\in \mathbb{R}\times {\mathbb{R}}^N$ and there exist $h_2>0$ and $p_1>2$ such that

$$|G_u(t,u)|\le h_2(1+{|u|}^{p_1-1}).$$

Then system (1.1) possesses infinitely many homoclinic solutions.

Remark 1.4 Obviously, all the conditions in Theorem 1.4 are more general than those in Theorem 1.2. Therefore, Theorem 1.4 is a complement of Theorem 1.2. On the other hand, there are many functions W satisfying our Theorem 1.4 and not satisfying Theorem 1.2. For example, the function

$$W(t,u)=F(t,u)+G(t,u),$$

(1.3)

where $F(t,u)={(\frac{1}{1+t^2})}^{\frac{1}{3}}{|u|}^{\frac{4}{3}}+{(\frac{1}{1+|t|})}^{\frac{1}{3}}{|u|}^{\frac{5}{3}}$ and $G(t,u)={|u|}^4+\frac{{\beta }_0}{4}{|u|}^2$. Thus, $F(t,u)$ is subquadratic and $G(t,u)$ is superquadratic. To the best of our knowledge, with the exception of [22, 23], the study of this case has received considerably less attention. Furthermore, it is easy to see that the function (1.3) does not satisfy Theorem 1.1 in [22].

Remark 1.5 In Theorem 1.4, there are many functions W satisfying (A₁) and not satisfying (A₂), for example, the function (1.3).

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proofs of Theorems 1.3 and 1.4.

2 Preliminaries

We will present some definitions and lemmas that will be used in the proofs of our results. Let

$$E=\{u\in H^1(\mathbb{R},{\mathbb{R}}^N):{\int }_{\mathbb{R}}[{|\dot{u}(t)|}^2+(L(t)u(t),u(t))]dt<+\mathrm{\infty }\}$$

equipped with the norm

$$\parallel u\parallel ={\left({\int }_{\mathbb{R}}[{|\dot{u}(t)|}^2+(L(t)u(t),u(t))]dt\right)}^{\frac{1}{2}},\mathrm{\forall }u\in X,$$

(2.1)

and the inner product

$$〈u,v〉={\int }_{\mathbb{R}}[(\dot{u}(t),\dot{v}(t))+(L(t)u(t),v(t))]dt,\mathrm{\forall }u,v\in X.$$

(2.2)

Then E is a Hilbert space with this inner product. Denote by $E^{\ast }$ its dual space with the associated operator norm ${\parallel \cdot \parallel }_{E^{\ast }}$. Note that E is continuously embedded in $L^p(\mathbb{R},{\mathbb{R}}^N)$ for all $p\in [2,+\mathrm{\infty }]$. Therefore, there exists a constant ${\delta }_p>0$ such that

$${\parallel u\parallel }_p\le {\delta }_p\parallel u\parallel ,\mathrm{\forall }u\in E,$$

(2.3)

where ${\parallel \cdot \parallel }_p$ denotes the usual norm on $L^p(\mathbb{R},{\mathbb{R}}^N)$.

Lemma 2.1 (see [13])

Suppose that L satisfies (C₁). Then the embedding of E in $L^2(\mathbb{R},{\mathbb{R}}^N)$ is compact.

Lemma 2.2 Suppose that (C₁), (C₂), (C₃) and (C₄) are satisfied. If $u_k\rightharpoonup u$ in E, then $W_u(t,u_k)\to W_u(t,u)$ in $L^2(\mathbb{R},{\mathbb{R}}^N)$.

Proof Assume that $u_k\rightharpoonup u$. In view of (C₂), (C₃) and (C₄), we get

$$\begin{array}{r}|W_u(t,u_k)-W_u(t,u)|\\ \le c_2(t)({|u_k|}^{\gamma -1}+{|u|}^{\gamma -1})+c_3(t)({|u_k|}^{\sigma -1}+{|u|}^{\sigma -1})\\ +|G_u(t,u_k)-G_u(t,u)|\\ \le c_2(t)({|u_k|}^{\gamma -1}+{|u|}^{\gamma -1})+c_3(t)({|u_k|}^{\sigma -1}+{|u|}^{\sigma -1})\\ +c(t)({|u_k|}^{\mu -1}+{|u|}^{\mu -1})+2a(t)\\ \le c_2(t)({|u_k-u|}^{\gamma -1}+2{|u|}^{\gamma -1})+c_3(t)({|u_k-u|}^{\sigma -1}+2{|u|}^{\sigma -1})\\ +c(t)[2^{\mu -2}{|u_k-u|}^{\mu -1}+(2^{\mu -2}+1){|u|}^{\mu -1}]+2a(t)\\ \le c_2(t)({|u_k-u|}^{\gamma -1}+2{|u|}^{\gamma -1})+c_3(t)({|u_k-u|}^{\sigma -1}+2{|u|}^{\sigma -1})\\ +2^{\mu -1}c(t)({|u_k-u|}^{\mu -1}+{|u|}^{\mu -1})+2a(t),\end{array}$$

(2.4)

which yields

$$\begin{array}{rcl}{|W_u(t,u_k)-W_u(t,u)|}^2& \le & 7c_2^2(t)({|u_k-u|}^{2\gamma -2}+4{|u|}^{2\gamma -2})\\ +7c_3^2(t)({|u_k-u|}^{2\sigma -2}+4{|u|}^{2\sigma -2})\\ +28a^2(t)+7\times 2^{2\mu -2}{c}^2(t)({|u_k-u|}^{2\mu -2}+{|u|}^{2\mu -2}).\end{array}$$

(2.5)

By virtue of (2.3), $u_k\rightharpoonup u$ and the Banach-Steinhaus Theorem, one has

$$\underset{k\in \mathbb{N}}{sup}{\parallel u_k\parallel }_{\mathrm{\infty }}\le M_1,{\parallel u\parallel }_{\mathrm{\infty }}\le M_1,$$

(2.6)

where $M_1>0$ is a constant. In view of $2\mu -4>0$, (2.5) and (2.6), we have

$$\begin{array}{r}{|W_u(t,u_k)-W_u(t,u)|}^2\\ \le 7c_2^2(t)({|u_k-u|}^{2\gamma -2}+4{|u|}^{2\gamma -2})\\ +7c_3^2(t)({|u_k-u|}^{2\sigma -2}+4{|u|}^{2\sigma -2})+28a^2(t)\\ +7\times 2^{2\mu -2}{c}^2(t)({|u_k-u|}^{2\mu -2}+{|u|}^{2\mu -2})\\ \le 7c_2^2(t)({|u_k-u|}^{2\gamma -2}+4{|u|}^{2\gamma -2})\\ +7c_3^2(t)({|u_k-u|}^{2\sigma -2}+4{|u|}^{2\sigma -2})+28a^2(t)\\ +7\times 2^{2\mu -2}{c}^2(t)[{(2M_1)}^{2\mu -4}{|u_k-u|}^2+{|u|}^{2\mu -2}].\end{array}$$

(2.7)

Since $u_k\rightharpoonup u$, then $u_k\to u$ in $L^2(\mathbb{R},{\mathbb{R}}^N)$, passing to a subsequence if necessary, we have

$${\parallel u_k-u\parallel }_2\to 0\text{as }k\to \mathrm{\infty }.$$

So it can be assumed that

$$\sum _{k=1}^{\mathrm{\infty }}{\parallel u_k-u\parallel }_2<+\mathrm{\infty },$$

which implies that $u_k(t)\to u(t)$ for almost every $t\in \mathbb{R}$ and

$$\sum _{k=1}^{\mathrm{\infty }}|u_k(t)-u(t)|=e(t)\in L^2(\mathbb{R},\mathbb{R}).$$

Then we have

$$\begin{array}{rcl}{|W_u(t,u_k)-W_u(t,u)|}^2& \le & 7c_2^2(t)({|e(t)|}^{2\gamma -2}+4{|u|}^{2\gamma -2})\\ +7c_3^2(t)({|e(t)|}^{2\sigma -2}+4{|u|}^{2\sigma -2})+28a^2(t)\\ +7\times 2^{2\mu -2}{c}^2(t)[{(2M_1)}^{2\mu -4}{|e(t)|}^2+{|u|}^{2\mu -2}].\end{array}$$

(2.8)

By (2.3), (2.8), $2\mu -2>2$ and the Hölder inequality, we obtain

$$\begin{array}{r}{\int }_{\mathbb{R}}{|W_u(t,u_k)-W_u(t,u)|}^{2}dt\\ \le 7{\int }_{\mathbb{R}}{c}_2^2(t)({|e(t)|}^{2\gamma -2}+4{|u|}^{2\gamma -2})dt\\ +7{\int }_{\mathbb{R}}{c}_3^2(t)({|e(t)|}^{2\sigma -2}+4{|u|}^{2\sigma -2})dt+28{\int }_{\mathbb{R}}{a}^2(t)dt\\ +7\times 2^{2\mu -2}{\parallel c\parallel }_{\mathrm{\infty }}^2{\int }_{\mathbb{R}}[{(2M_1)}^{2\mu -4}{|e(t)|}^2+{|u|}^{2\mu -2}]dt\\ \le 7{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}^2({\parallel e\parallel }_2^{2\gamma -2}+4{\parallel u\parallel }_2^{2\gamma -2})+28{\parallel a\parallel }_2^2\\ +7{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}^2({\parallel e\parallel }_2^{2\sigma -2}+4{\parallel u\parallel }_2^{2\sigma -2})\\ +7\times 2^{2\mu -2}{\parallel c\parallel }_{\mathrm{\infty }}^2[{(2M_1)}^{2\mu -4}{\parallel e\parallel }_2^2+{\parallel u\parallel }_{2\mu -2}^{2\mu -2}]\\ \le 7{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}^2({\parallel e\parallel }_2^{2\gamma -2}+4{\delta }_2^{2\gamma -2}{\parallel u\parallel }^{2\gamma -2})+28{\parallel a\parallel }_2^2\\ +7{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}^2({\parallel e\parallel }_2^{2\sigma -2}+4{\delta }_2^{2\sigma -2}{\parallel u\parallel }^{2\sigma -2})\\ +7\times 2^{2\mu -2}{\parallel c\parallel }_{\mathrm{\infty }}^2[{(2M_1)}^{2\mu -4}{\parallel e\parallel }_2^2+{\delta }_{2\mu -2}^{2\mu -2}{\parallel u\parallel }^{2\mu -2}].\end{array}$$

By using the Lebesgue dominated convergence theorem, the lemma is proved. □

Remark 2.1 Suppose that the condition (C₄) is replaced by conditions (C₇) and (C₈), then we can obtain the same conclusion.

Define the functional Ψ on E by

$$\begin{array}{rcl}\mathrm{\Psi }(u)& =& \frac{1}{2}{\int }_{\mathbb{R}}[{|\dot{u}(t)|}^2+(L(t)u(t),u(t))]dt-{\int }_{\mathbb{R}}W(t,u(t))dt\\ =& \frac{1}{2}{\parallel u\parallel }^2-\phi (u),\end{array}$$

(2.9)

where $\phi (u)={\int }_{\mathbb{R}}W(t,u(t))dt$.

Lemma 2.3 Under conditions (C₁)-(C₄), we have

$$\begin{array}{rl}{\mathrm{\Psi }}^{\prime }(u)v& ={\int }_{\mathbb{R}}[(\dot{u}(t),\dot{v}(t))+(L(t)u(t),v(t))]dt-{\int }_{\mathbb{R}}(W_u(t,u(t)),v(t))dt\\ ={\int }_{\mathbb{R}}[(\dot{u}(t),\dot{v}(t))+(L(t)u(t),v(t))]dt-{\phi }^{\prime }(u)v\end{array}$$

(2.10)

for any $u,v\in E$, which yields

$${\mathrm{\Psi }}^{\prime }(u)u={\parallel u\parallel }^2-{\int }_{\mathbb{R}}(W_u(t,u(t)),u(t))dt.$$

(2.11)

Moreover, $\mathrm{\Psi }\in C^1(E,\mathbb{R})$, ${\phi }^{\prime }:E\to E^{\ast }$ is compact and any critical point of Ψ on E is a classical solution for system (1.1) satisfying $u\in C^2(\mathbb{R},{\mathbb{R}}^N)$, $u(t)\to 0$ and $\dot{u}(t)\to 0$ as $|t|\to \mathrm{\infty }$.

Proof We first show that $\mathrm{\Psi }:E\to \mathbb{R}$. It follows from (C₂), (C₃), (C₄), (2.3), $\mu >2$ and the Hölder inequality that

$$\begin{array}{rcl}0& \le & {\int }_{\mathbb{R}}W(t,u(t))dt\\ \le & {\int }_{\mathbb{R}}(\frac{1}{\gamma }{c}_2(t){|u(t)|}^{\gamma }+\frac{1}{\sigma }{c}_3(t){|u(t)|}^{\sigma })dt+{\int }_{\mathbb{R}}(a(t)|u(t)|+c(t){|u(t)|}^{\mu })dt\\ \le & \frac{1}{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u\parallel }_2^{\gamma }+\frac{1}{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u\parallel }_2^{\sigma }+{\parallel a\parallel }_2{\parallel u\parallel }_2+{\parallel c\parallel }_{\mathrm{\infty }}{\parallel u\parallel }_{\mu }^{\mu }\\ \le & \frac{1}{\gamma }{\delta }_2^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u\parallel }^{\gamma }+\frac{1}{\sigma }{\delta }_2^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u\parallel }^{\sigma }+{\delta }_2{\parallel a\parallel }_2\parallel u\parallel +{\parallel c\parallel }_{\mathrm{\infty }}{\delta }_{\mu }^{\mu }{\parallel u\parallel }^{\mu }.\end{array}$$

Next, we prove that $\mathrm{\Psi }\in C^1(E,\mathbb{R})$. It is sufficient to show that $\phi \in C^1(E,\mathbb{R})$. At first, we will see that

$${\phi }^{\prime }(u)v={\int }_{\mathbb{R}}(W_u(t,u(t)),v(t))dt$$

(2.12)

for any $u,v\in E$. For any given $u\in E$, let us define $\mathrm{\Phi }(u):E\to \mathbb{R}$ as follows:

$$\mathrm{\Phi }(u)v={\int }_{\mathbb{R}}(W_u(t,u(t)),v(t))dt,\mathrm{\forall }v\in E.$$

It is easy to see that $\mathrm{\Phi }(u)$ is linear. In the following we show that $\mathrm{\Phi }(u)$ is bounded. In fact, for any $u\in E$, by (C₂), (C₃), (C₄), (2.3) and the Hölder inequality, we have

$$\begin{array}{rcl}|\mathrm{\Phi }(u)v|& =& \left|{\int }_{\mathbb{R}}(W_u(t,u(t)),v(t))dt\right|\\ \le & {\int }_{\mathbb{R}}{c}_2(t){|u(t)|}^{\gamma -1}|v(t)|dt+{\int }_{\mathbb{R}}{c}_3(t){|u(t)|}^{\sigma -1}|v(t)|dt\\ +{\int }_{\mathbb{R}}a(t)|v(t)|dt+{\int }_{\mathbb{R}}c(t){|u(t)|}^{\mu -1}|v(t)|dt\\ \le & {({\int }_{\mathbb{R}}{c}_2^2(t){|u(t)|}^{2\gamma -2}dt)}^{\frac{1}{2}}{\parallel v\parallel }_2+{({\int }_{\mathbb{R}}{c}_3^2(t){|u(t)|}^{2\sigma -2}dt)}^{\frac{1}{2}}{\parallel v\parallel }_2\\ +{\parallel a\parallel }_2{\parallel v\parallel }_2+{\parallel c\parallel }_{\mathrm{\infty }}{\left({\int }_{\mathbb{R}}{|u(t)|}^{2\mu -2}dt\right)}^{\frac{1}{2}}{\parallel v\parallel }_2\\ \le & ({\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u\parallel }_2^{\gamma -1}+{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u\parallel }_2^{\sigma -1}+{\parallel a\parallel }_2+{\parallel c\parallel }_{\mathrm{\infty }}{\parallel u\parallel }_{2\mu -2}^{\mu -1}){\parallel v\parallel }_2\\ \le & {\delta }_2({\delta }_2^{\gamma -1}{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u\parallel }^{\gamma -1}+{\delta }_2^{\sigma -1}{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u\parallel }^{\sigma -1}\\ +{\parallel a\parallel }_2+{\parallel c\parallel }_{\mathrm{\infty }}{\delta }_{2\mu -2}^{\mu -1}{\parallel u\parallel }^{\mu -1})\parallel v\parallel .\end{array}$$

Moreover, for any $u,v\in E$, by the mean value theorem, we have

$${\int }_{\mathbb{R}}[W(t,u(t)+v(t))-W(t,u(t))]dt={\int }_{\mathbb{R}}(W_u(t,u(t)+\vartheta (t)v(t)),v(t))dt,$$

where $\vartheta (t)\in (0,1)$. Thus, by Lemma 2.2 and the Hölder inequality, one has

$${\int }_{\mathbb{R}}(W_u(t,u(t)+\vartheta (t)v(t))-W_u(t,u(t)),v(t))dt\to 0\text{as }v\to 0.$$

Suppose that $u\to u_0$ in E and note that

$${\phi }^{\prime }(u)v-{\phi }^{\prime }(u_0)v={\int }_{\mathbb{R}}(W_u(t,u(t))-W_u(t,u_0(t)),v(t))dt.$$

Combining Lemma 2.2 and the Hölder inequality, we have

$${\phi }^{\prime }(u)v-{\phi }^{\prime }(u_0)v\to 0\text{as }u\to u_0.$$

So ${\phi }^{\prime }$ is continuous and $\mathrm{\Psi }\in C^1(E,\mathbb{R})$. Let $u_k\rightharpoonup u$ in E, we get

$$\begin{array}{rcl}{\parallel {\phi }^{\prime }(u_k)-{\phi }^{\prime }(u)\parallel }_{E^{\ast }}& =& \underset{\parallel v\parallel =1}{sup}\parallel ({\phi }^{\prime }(u_k)-{\phi }^{\prime }(u))v\parallel \\ =& \underset{\parallel v\parallel =1}{sup}\left|{\int }_{\mathbb{R}}(W_u(t,u_k(t))-W_u(t,u(t)),v(t))dt\right|\\ \le & \underset{\parallel v\parallel =1}{sup}{\left({\int }_{\mathbb{R}}{|W_u(t,u_k(t))-W_u(t,u(t))|}^{2}dt\right)}^{\frac{1}{2}}{\parallel v\parallel }_2\\ \le & {\delta }_2{\left({\int }_{\mathbb{R}}{|W_u(t,u_k(t))-W_u(t,u(t))|}^{2}dt\right)}^{\frac{1}{2}}\to 0\end{array}$$

as $k\to \mathrm{\infty }$. Consequently, ${\phi }^{\prime }$ is weakly continuous. Therefore, ${\phi }^{\prime }$ is compact by the weakly continuity of ${\phi }^{\prime }$ since E is a Hilbert space.

Finally, as in the discussion in Lemma 3.1 of [29], we find that the critical points of Ψ are classical solutions of system (1.1) satisfying $u\in C^2(\mathbb{R},{\mathbb{R}}^N)$, $u(t)\to 0$ and $\dot{u}(t)\to 0$ as $|t|\to \mathrm{\infty }$. The proof is complete. □

Remark 2.2 If condition (C₄) is replaced by conditions (C₇) and (C₈), then we can obtain the same conclusion.

In the next section we shall prove our results applying the dual fountain theorem obtained in [35] (see also Proposition 2.1 of [24]). Assume that E be a Banach space with the norm $\parallel \cdot \parallel$ and $E=\overline{{⨁}_{j\in \mathbb{N}}{X}_j}$, where $X_j$ is a finite-dimensional subspace of E. For each $k\in \mathbb{N}$, let $Y_k={⨁}_{j=0}^k{X}_j$, $Z_k=\overline{{⨁}_{j=k}^{\mathrm{\infty }}{X}_j}$. The functional Ψ is said to satisfy the ${(PS)}^{\ast }$ condition if for any sequence $\{u_j\}$ for which $\{\mathrm{\Psi }(u_j)\}$ is bounded, $u_j\in Y_{k_j}$ for some $k_j$ with $k_j\to \mathrm{\infty }$ and ${(\mathrm{\Psi }{|}_{Y_{k_j}})}^{\prime }(u_j)\to 0$ as $j\to \mathrm{\infty }$ has a subsequence converging to a critical point of Ψ.

Theorem 2.1 Suppose that the functional $\mathrm{\Psi }\in C^1(E,\mathbb{R})$ is even and satisfies the ${(PS)}^{\ast }$ condition. Assume that for each sufficiently large $k\in \mathbb{N}$, there exist ${\rho }_k>r_k>0$ such that

(H₁) $a_k:={inf}_{u\in Z_k,\parallel u\parallel ={\rho }_k}\mathrm{\Psi }(u)\ge 0$.

(H₂) $b_k:={max}_{u\in Y_k,\parallel u\parallel =r_k}\mathrm{\Psi }(u)<0$.

(H₃) $d_k:={inf}_{u\in Z_k,\parallel u\parallel \le {\rho }_k}\mathrm{\Psi }(u)\to 0$ as $k\to \mathrm{\infty }$.

Then Ψ has a sequence of negative critical values converging to 0.

3 Proof of Theorem 1.3 and 1.4

Now we give the proof of Theorem 1.3.

Proof of Theorem 1.3 We choose a completely orthonormal basis $\{e_j\}$ of X and define $X_j:=\mathbb{R}{e}_j$, then $Z_k$ and $Y_k$ can be defined as that in Section 2. By (C₂) and Lemma 2.3, we see that $\mathrm{\Psi }\in C^1(E,\mathbb{R})$ is even. In the following, we will check that all conditions in Theorem 2.1 are satisfied.

Step 1. We prove that Ψ satisfies the ${(PS)}^{\ast }$ condition. Let $\{u_j\}$ be a ${(PS)}^{\ast }$ sequence, that is, $\{\mathrm{\Psi }(u_j)\}$ is bounded, $u_j\in Y_{k_j}$ for some $k_j$ with $k_j\to \mathrm{\infty }$ and ${(\mathrm{\Psi }{|}_{Y_{k_j}})}^{\prime }(u_j)\to 0$ as $j\to \mathrm{\infty }$. Now we show that $\{u_j\}$ is bounded in E. By virtue of (C₂), (C₃) and (C₅), for j large enough, we have

$$\begin{array}{rcl}\rho M_2+M_3\parallel u_j\parallel & \ge & \rho \mathrm{\Psi }(u_j)-{\mathrm{\Psi }}^{\prime }(u_j)u_j\\ =& (\frac{\rho }{2}-1){\parallel u_j\parallel }^2+{\int }_{\mathbb{R}}[(W_u(t,u_j),u_j)-\rho W(t,u_j)]dt\\ =& (\frac{\rho }{2}-1){\parallel u_j\parallel }^2+{\int }_{\mathbb{R}}[(F_u(t,u_j),u_j)-\rho F(t,u_j)]dt\\ +{\int }_{\mathbb{R}}[(G_u(t,u_j),u_j)-\rho G(t,u_j)]dt\\ \ge & (\frac{\rho }{2}-1){\parallel u_j\parallel }^2-(\rho +1){\int }_{\mathbb{R}}(c_2(t){|u_j(t)|}^{\gamma }+c_3{|u_j(t)|}^{\sigma })dt\\ -{\int }_{\mathbb{R}}h(t){|u_j(t)|}^{\delta }dt\\ \ge & (\frac{\rho }{2}-1){\parallel u_j\parallel }^2-(\rho +1)({\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u_j\parallel }_2^{\gamma }+{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u_j\parallel }_2^{\sigma })\\ -{\parallel h\parallel }_{\frac{2}{2-\delta }}{\parallel u_j\parallel }_2^{\delta }\\ \ge & (\frac{\rho }{2}-1){\parallel u_j\parallel }^2-(\rho +1)({\delta }_2^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u_j\parallel }^{\gamma }+{\delta }_2^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u_j\parallel }^{\sigma })\\ -{\delta }_2^{\delta }{\parallel h\parallel }_{\frac{2}{2-\delta }}{\parallel u_j\parallel }^{\delta }\end{array}$$

(3.1)

for some $M_2>0$, $M_3>0$. Since $\rho >2$ and $\gamma ,\sigma ,\delta <2$, it follows that $\{u_j\}$ is bounded in E.

From the reflexivity of E, we may extract a weakly convergent subsequence, which, for simplicity, we call $\{u_j\}$, $u_j\rightharpoonup u_1$ in E. In view of the Riesz representation theorem, ${(\mathrm{\Psi }{|}_{Y_{k_j}})}^{\prime }:Y_{k_j}\to Y_{k_j}^{\ast }$ and ${\phi }^{\prime }:E\to E^{\ast }$ can be viewed as ${(\mathrm{\Psi }{|}_{Y_{k_j}})}^{\prime }:Y_{k_j}\to Y_{k_j}$ and ${\phi }^{\prime }:E\to E$, respectively, where $Y_{k_j}^{\ast }$ is the dual space of $Y_{k_j}$. Note that

$${(\mathrm{\Psi }{|}_{Y_{k_j}})}^{\prime }(u_j)=u_j-P_{k_j}{\phi }^{\prime }(u_j),\mathrm{\forall }j\in \mathbb{N},$$

(3.2)

where $P_{k_j}:E\to Y_{k_j}$ is the orthogonal projection for all $j\in \mathbb{N}$. That is,

$$u_j={(\mathrm{\Psi }{|}_{Y_{k_j}})}^{\prime }(u_j)+P_{k_j}{\phi }^{\prime }(u_j),\mathrm{\forall }j\in \mathbb{N}.$$

(3.3)

Due to the compactness of ${\phi }^{\prime }$ and $u_j\rightharpoonup u_1$, the right hand side of (3.3) converges strongly in E and hence $u_j\to u_1$ in E.

Step 2. We verify condition (H₁) in Theorem 2.1. Set ${\beta }_k={sup}_{u\in Z_k,\parallel u\parallel =1}{\parallel u\parallel }_2$, then ${\beta }_k\to 0$ as $k\to \mathrm{\infty }$ since E is compactly embedded into $L^2(\mathbb{R},{\mathbb{R}}^N)$. By (C₂), (C₃) and (C₄), we have

$$\begin{array}{rcl}\mathrm{\Psi }(u)& =& \frac{1}{2}{\parallel u\parallel }^2-{\int }_{\mathbb{R}}W(t,u(t))dt\\ \ge & \frac{1}{2}{\parallel u\parallel }^2-{\int }_{\mathbb{R}}(c_2(t){|u(t)|}^{\gamma }+c_3(t){|u(t)|}^{\sigma })dt\\ -{\int }_{\mathbb{R}}(a(t)|u(t)|+c(t){|u(t)|}^{\mu })dt\\ \ge & \frac{1}{2}{\parallel u\parallel }^2-{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u\parallel }_2^{\gamma }-{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u\parallel }_2^{\sigma }-{\parallel a\parallel }_2{\parallel u\parallel }_2-{\parallel c\parallel }_{\mathrm{\infty }}{\parallel u\parallel }_{\mu }^{\mu }\\ \ge & \frac{1}{2}{\parallel u\parallel }^2-{\beta }_k^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u\parallel }^{\gamma }-{\beta }_k^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u\parallel }^{\sigma }-{\beta }_k{\parallel a\parallel }_2\parallel u\parallel \\ -{\delta }_{\mu }^{\mu }{\parallel c\parallel }_{\mathrm{\infty }}{\parallel u\parallel }^{\mu }.\end{array}$$

(3.4)

In view of (3.4), $\mu >2$ and $\gamma ,\sigma >1$, one has

$$\mathrm{\Psi }(u)\ge \frac{1}{4}{\parallel u\parallel }^2-({\beta }_k^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}+{\beta }_k^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}+{\beta }_k{\parallel a\parallel }_2)\parallel u\parallel$$

(3.5)

for $\parallel u\parallel$ small enough. Let ${\rho }_k=8({\beta }_k^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}+{\beta }_k^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}+{\beta }_k{\parallel a\parallel }_2)$, it is easy to see that ${\rho }_k\to 0$ as $k\to \mathrm{\infty }$. Thus, for each sufficiently large $k\in \mathbb{N}$, by (3.5), we get

$$a_k\ge \frac{1}{8}{\rho }_k^2>0.$$

Step 3. We verify condition (H₃) in Theorem 2.1. By (3.5), for any $u\in Z_k$ with $\parallel u\parallel \le {\rho }_k$, we have

$$\mathrm{\Psi }(u)\ge -({\beta }_k^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}+{\beta }_k^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}+{\beta }_k{\parallel a\parallel }_2)\parallel u\parallel .$$

(3.6)

Therefore, by (C₂), we obtain

$$0\ge d_k\ge -({\beta }_k^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}+{\beta }_k^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}+{\beta }_k{\parallel a\parallel }_2)\parallel u\parallel .$$

(3.7)

Since ${\beta }_k,{\rho }_k\to 0$ as $k\to \mathrm{\infty }$, one has

$$d_k=\underset{u\in Z_k,\parallel u\parallel \le {\rho }_k}{inf}\mathrm{\Psi }(u)\to 0\text{as }k\to \mathrm{\infty }.$$

Step 4. We verify condition (H₂) in Theorem 2.1. Firstly, we claim that there exists $\epsilon >0$ such that

$$meas\{t\in \mathbb{R}:c_1(t){|u(t)|}^{\gamma }\ge \epsilon {\parallel u\parallel }^{\gamma }\}\ge \epsilon ,\mathrm{\forall }u\in Y_k\setminus \{0\}.$$

(3.8)

If not, there exists a sequence $\{u_n\}\subset Y_k$ with $\parallel u_n\parallel =1$ such that

$$meas\{t\in \mathbb{R}:c_1(t){|u_n(t)|}^{\gamma }\ge \frac{1}{n}\}\le \frac{1}{n}.$$

(3.9)

Since $dim{Y}_k<\mathrm{\infty }$, it follows from the compactness of the unit sphere of $Y_k$ that there exists a subsequence, say $\{u_n\}$, such that $u_n$ converges to some $u_0$ in $Y_k$. Hence, we have $\parallel u_0\parallel =1$. Since all norms are equivalent in the finite-dimensional space, we have $u_n\to u_0$ in $L^2(\mathbb{R},{\mathbb{R}}^N)$. By the Hölder inequality, one has

$${\int }_{\mathbb{R}}{c}_1(t){|u_n-u_0|}^{\gamma }dt\le {\parallel c_1\parallel }_{\frac{2}{2-\gamma }}{\left({\int }_{\mathbb{R}}{|u_n-u_0|}^{2}dt\right)}^{\frac{\gamma }{2}}\to 0\text{as }n\to \mathrm{\infty }.$$

(3.10)

Thus there exist ${ϵ}_1,{ϵ}_2>0$ such that

$$meas\{t\in \mathbb{R}:c_1(t){|u_0(t)|}^{\gamma }\ge {ϵ}_1\}\ge {ϵ}_2.$$

(3.11)

In fact, if not, we have

$$meas\{t\in \mathbb{R}:c_1(t){|u_0(t)|}^{\gamma }\ge \frac{1}{n}\}=0,$$

(3.12)

for all positive integers n, which implies that

$${\int }_{\mathbb{R}}{c}_1(t){|u_0(t)|}^{\gamma +2}dt<\frac{1}{n}{\parallel u_0\parallel }_2^2\le \frac{{\delta }_2^2}{n}{\parallel u_0\parallel }^2=\frac{{\delta }_2^2}{n}\to 0$$

as $n\to \mathrm{\infty }$. Hence $u_0=0$, which contradicts $\parallel u_0\parallel =1$. Therefore, (3.11) holds. Thus, define

$${\mathrm{\Omega }}_0=\{t\in \mathbb{R}:c_1(t){|u_0(t)|}^{\gamma }\ge {ϵ}_1\},{\mathrm{\Omega }}_n=\{t\in \mathbb{R}:c_1(t){|u_n(t)|}^{\gamma }<\frac{1}{n}\}$$

and ${\mathrm{\Omega }}_n^c=\mathbb{R}\setminus {\mathrm{\Omega }}_n=\{t\in \mathbb{R}:c_1(t){|u_n(t)|}^{\gamma }\ge \frac{1}{n}\}$. Combining (3.9) and (3.11), we have

$$\begin{array}{rcl}meas({\mathrm{\Omega }}_n\cap {\mathrm{\Omega }}_0)& =& meas({\mathrm{\Omega }}_0\setminus {\mathrm{\Omega }}_n^c\cap {\mathrm{\Omega }}_0)\\ \ge & meas({\mathrm{\Omega }}_0)-meas({\mathrm{\Omega }}_n^c\cap {\mathrm{\Omega }}_0)\\ \ge & {ϵ}_2-\frac{1}{n}\end{array}$$

for all positive integers n. Let n be large enough such that ${ϵ}_2-\frac{1}{n}\ge \frac{1}{2}{ϵ}_2$ and $\frac{1}{2^{\gamma -1}}{ϵ}_1-\frac{1}{n}\ge \frac{1}{2^{\gamma }}{ϵ}_1$. Then we have

$$\begin{array}{rcl}{\int }_{\mathbb{R}}{c}_1(t){|u_n-u_0|}^{\gamma }dt& \ge & {\int }_{{\mathrm{\Omega }}_n\cap {\mathrm{\Omega }}_0}{c}_1(t){|u_n-u_0|}^{\gamma }dt\\ \ge & \frac{1}{2^{\gamma -1}}{\int }_{{\mathrm{\Omega }}_n\cap {\mathrm{\Omega }}_0}{c}_1(t){|u_0|}^{\gamma }dt-{\int }_{{\mathrm{\Omega }}_n\cap {\mathrm{\Omega }}_0}{c}_1(t){|u_n|}^{\gamma }dt\\ \ge & (\frac{1}{2^{\gamma -1}}{ϵ}_1-\frac{1}{n})meas({\mathrm{\Omega }}_n\cap {\mathrm{\Omega }}_0)\\ \ge & \frac{{ϵ}_1{ϵ}_2}{2^{\gamma +1}}\end{array}$$

for all large n, which is a contradiction to (3.10). Therefore, (3.8) holds. For the ε given in (3.8), let

$${\mathrm{\Omega }}_u=\{t\in \mathbb{R}:c_1(t){|u(t)|}^{\gamma }\ge \epsilon {\parallel u\parallel }^{\gamma }\},\mathrm{\forall }u\in Y_k\setminus \{0\}.$$

(3.13)

By (3.8), we obtain

$$meas({\mathrm{\Omega }}_u)\ge \epsilon ,\mathrm{\forall }u\in Y_k\setminus \{0\}.$$

(3.14)

For any $u_k\in Y_k$, by (C₂), (C₃), (C₄), (3.13) and (3.14), we have

$$\begin{array}{rcl}\mathrm{\Psi }(u)& =& \frac{1}{2}{\parallel u\parallel }^2-{\int }_{\mathbb{R}}W(t,u(t))dt\\ \le & \frac{1}{2}{\parallel u\parallel }^2-\frac{1}{\gamma }{\int }_{\mathbb{R}}{c}_1(t){|u(t)|}^{\gamma }dt-{\int }_{\mathbb{R}}G(t,u(t))dt\\ \le & \frac{1}{2}{\parallel u\parallel }^2-\frac{1}{\gamma }{\int }_{\mathbb{R}}{c}_1(t){|u(t)|}^{\gamma }dt\\ \le & \frac{1}{2}{\parallel u\parallel }^2-\frac{1}{\gamma }{\int }_{{\mathrm{\Omega }}_u}{c}_1(t){|u(t)|}^{\gamma }dt\\ \le & \frac{1}{2}{\parallel u\parallel }^2-\frac{\epsilon }{\gamma }{\parallel u\parallel }^{\gamma }meas({\mathrm{\Omega }}_u)\\ \le & \frac{1}{2}{\parallel u\parallel }^2-\frac{{\epsilon }^2}{\gamma }{\parallel u\parallel }^{\gamma }.\end{array}$$

Choose $0<r_k<min\{{\rho }_k,{({\epsilon }^2/\gamma )}^{\frac{1}{2-\gamma }}\}$. Direct computation shows that

$$b_k\le -\frac{r_k^2}{2}<0,\mathrm{\forall }k\in \mathbb{N}.$$

Thus, by Theorem 2.1, Ψ has infinitely many nontrivial critical points, that is, system (1.1) possesses infinitely many homoclinic solutions. □

Now we give the proof of Theorem 1.4.

Proof of Theorem 1.4 Step 1. We prove that Ψ satisfies the ${(PS)}^{\ast }$ condition. Let $\{u_j\}$ be a ${(PS)}^{\ast }$ sequence, that is, $\{\mathrm{\Psi }(u_j)\}$ is bounded, $u_j\in Y_{k_j}$ for some $k_j$ with $k_j\to \mathrm{\infty }$ and ${(\mathrm{\Psi }{|}_{Y_{k_j}})}^{\prime }(u_j)\to 0$ as $j\to \mathrm{\infty }$. Now we show that $\{u_j\}$ is bounded in E. In view of (C₁), (C₂), (C₃) and (C₆), for j large enough, we obtain

$$\begin{array}{rcl}\lambda M_4+M_5\parallel u_j\parallel & \ge & \lambda \mathrm{\Psi }(u_j)-{\mathrm{\Psi }}^{\prime }(u_j)u_j\\ =& (\frac{\lambda }{2}-1){\parallel u_j\parallel }^2+{\int }_{\mathbb{R}}[(W_u(t,u_j),u_j)-\lambda W(t,u_j)]dt\\ =& (\frac{\lambda }{2}-1){\parallel u_j\parallel }^2+{\int }_{\mathbb{R}}[(F_u(t,u_j),u_j)-\lambda F(t,u_j)]dt\\ +{\int }_{\mathbb{R}}[(G_u(t,u_j),u_j)-\lambda G(t,u_j)]dt\\ \ge & (\frac{\lambda }{2}-1){\parallel u_j\parallel }^2-(\lambda +1){\int }_{\mathbb{R}}(c_2(t){|u_j(t)|}^{\gamma }+c_3{|u_j(t)|}^{\sigma })dt\\ -h_1{\int }_{\mathbb{R}}{|u_j(t)|}^{2}dt\\ \ge & (\frac{\lambda }{2}-1){\parallel u_j\parallel }^2-(\lambda +1)({\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u_j\parallel }_2^{\gamma }+{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u_j\parallel }_2^{\sigma })\\ -\frac{h_1}{{\beta }_0}{\int }_{\mathbb{R}}(L(t)u_j(t),u_j(t))dt\\ \ge & (\frac{\lambda -2}{2}-\frac{h_1}{{\beta }_0}){\parallel u_j\parallel }^2\\ -(\lambda +1)({\delta }_2^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u_j\parallel }^{\gamma }+{\delta }_2^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u_j\parallel }^{\sigma })\end{array}$$

(3.15)

for some $M_4>0$, $M_5>0$. Since $0\le h_1<\frac{{\beta }_0(\lambda -2)}{2}$ and $1<\gamma ,\sigma <2$, it follows that $\{u_j\}$ is bounded in E. In the following, the proof of the ${(PS)}^{\ast }$ condition is the same as that in Theorem 1.1, and we omit it here.

Step 2. We verify condition (H₁) in Theorem 2.1. Set ${\beta }_k={sup}_{u\in Z_k,\parallel u\parallel =1}{\parallel u\parallel }_2$, then ${\beta }_k\to 0$ as $k\to \mathrm{\infty }$ since E is compactly embedded into $L^2(\mathbb{R},{\mathbb{R}}^N)$. By (C₈), we have

$$|G(t,u)|\le h_2(|u|+{|u|}^{p_1}).$$

(3.16)

It follows from (C₇) that there exists ${\delta }_0>0$ such that

$$|G(t,u)|\le \frac{\theta }{2}{|u|}^2,\mathrm{\forall }|u|\le {\delta }_0,\mathrm{\forall }t\in \mathbb{R}.$$

(3.17)

Combining (3.16) and (3.17), we get

$$|G(t,u)|\le \frac{\theta }{2}{|u|}^2+h_3{|u|}^{p_1},\mathrm{\forall }(t,u)\in \mathbb{R}\times {\mathbb{R}}^N,$$

(3.18)

where $h_3=h_2(1+{\delta }_0^{1-p_1})$. By (C₁), (C₂), (C₃) and (3.18), we have

$$\begin{array}{rcl}\mathrm{\Psi }(u)& =& \frac{1}{2}{\parallel u\parallel }^2-{\int }_{\mathbb{R}}W(t,u(t))dt\\ \ge & \frac{1}{2}{\parallel u\parallel }^2-{\int }_{\mathbb{R}}(c_2(t){|u(t)|}^{\gamma }+c_3(t){|u(t)|}^{\sigma })dt-{\int }_{\mathbb{R}}(\frac{\theta }{2}{|u(t)|}^2+h_3{|u(t)|}^{p_1})dt\\ \ge & \frac{1}{2}{\parallel u\parallel }^2-{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u\parallel }_2^{\gamma }-{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u\parallel }_2^{\sigma }-h_3{\parallel u\parallel }_{p_1}^{p_1}\\ -\frac{\theta }{2{\beta }_0}{\int }_{\mathbb{R}}(L(t)u_j(t),u_j(t))dt\\ \ge & (\frac{1}{2}-\frac{\theta }{2{\beta }_0}){\parallel u\parallel }^2-{\beta }_k^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}{\parallel u\parallel }^{\gamma }-{\beta }_k^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }}{\parallel u\parallel }^{\sigma }-h_3{\delta }_{p_1}^{p_1}{\parallel u\parallel }^{p_1}.\end{array}$$

(3.19)

Take ${\theta }_0=\frac{1}{2}-\frac{\theta }{2{\beta }_0}$, by (C₇), we obtain ${\theta }_0>0$. By virtue of (3.19), $p_1>2$ and $\gamma ,\sigma >1$, one has

$$\mathrm{\Psi }(u)\ge \frac{{\theta }_0}{2}{\parallel u\parallel }^2-({\beta }_k^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}+{\beta }_k^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }})\parallel u\parallel$$

(3.20)

for $\parallel u\parallel$ small enough. Let ${\rho }_k=\frac{{\theta }_0}{4}({\beta }_k^{\gamma }{\parallel c_2\parallel }_{\frac{2}{2-\gamma }}+{\beta }_k^{\sigma }{\parallel c_3\parallel }_{\frac{2}{2-\sigma }})$, it is easy to see that ${\rho }_k\to 0$ as $k\to \mathrm{\infty }$. Thus, for each sufficiently large $k\in \mathbb{N}$, by (3.20), we get

$$a_k\ge \frac{{\theta }_0}{4}{\rho }_k^2>0.$$

Step 3. We verify condition (H₃) in Theorem 2.1. The proof is similar to the Step 3 in the proof of Theorem 1.3, and we omit it.

Step 4. We verify condition (H₂) in Theorem 2.1. The proof is the same as that the Step 4 in the proof of Theorem 1.3, and we omit it here.

Thus, by Theorem 2.1, Ψ has infinitely many nontrivial critical points, that is, system (1.1) possesses infinitely many homoclinic solutions. □