New conditions on homoclinic solutions for a subquadratic second order Hamiltonian system

Abstract

In this paper, we deal with the second-order Hamiltonian system

$$ (*)\quad\ddot{u}-L(t)u+\nabla W(t, u)=0. $$

We establish some criteria which guarantee that the above system has at least one or infinitely many homoclinic solutions under the assumption that $W(t, x)$ is subquadratic at infinity and $L(t)$ is a real symmetric matrix and satisfies

$$\liminf_{|t|\to+\infty} \Bigl[|t|^{\nu-2}\inf _{|x|=1}\bigl(L(t)x, x\bigr) \Bigr] >0 $$

for some constant $\nu<2$. In particular, $L(t)$ and $W(t, x)$ are allowed to be sign-changing.

1 Introduction

Consider the second-order Hamiltonian system

$$ \ddot{u}-L(t)u+\nabla W(t, u)=0, $$

(1.1)

where $t\in{ \mathbb {R}}$, $u\in{ \mathbb {R}}^{N}$, $L\in C({\mathbb {R}}, {\mathbb {R}}^{N\times N})$ is a symmetric matrix-valued function, $W\in C^{1}(\mathbb {R}\times{{ \mathbb {R}}}^{N}, {\mathbb {R}})$ and $\nabla W(t, x)=\nabla_{x} W(t, x)$. As usual [1], we say that a solution $u(t)$ of system (1.1) is homoclinic (to 0) if $u(t)\rightarrow0$ as $t\to\pm\infty$. In addition, if $u(t)\not\equiv0$ then $u(t)$ is called a nontrivial homoclinic solution.

The existence and multiplicity of nontrivial homoclinic solutions for problem (1.1) have been extensively investigated in the literature with the aid of critical point theory and variational methods (see, for example, [2–17]). Most of them treat the case where $W(t, x)$ is superquadratic as $|x|\to\infty$.

Compared to the superquadratic case, as far as the authors are aware, there are a few papers [17–20] concerning the case where $W(t, x)$ has subquadratic growth at infinity. In these papers, since $L(t)$ is positive definite, the energy functional associated with system (1.1) is bounded from below, techniques based on the genus properties have been well applied. In particular, Clark’s theorem is an effective tool to prove the existence and multiplicity of homoclinic solutions for system (1.1). However, if $L(t)$ is not global positive definite on $\mathbb {R}$, the problem is far more difficult as 0 is a saddle point rather than a local minimum of the energy functional, which is strongly indefinite and it is not easy to prove the boundedness of the Palais-Smale sequence.

In [21], Ding studied the existence of homoclinic solutions of system (1.1) under the case when $L(t)$ is not global positive definite on $\mathbb {R}$ and $W(t, x)$ is subquadratic at infinity. He obtained the following result.

Theorem A

([21])

Assume that L and W satisfy the following conditions:

• (A1) There exists a constant $\nu<2$ such that

  $$|t|^{\nu-2}\inf_{|x|=1}\bigl(L(t)x, x\bigr)\rightarrow+ \infty \quad\textit{as } |t|\to+\infty; $$

• (A2) $0< \inf_{t\in \mathbb {R}, |x|=1}W(t, x)\le\sup_{t\in \mathbb {R}, |x|=1}W(t, x)<+\infty$;

• (A3) There exists a constant μ with $1<\mu\in((4-\nu )/(3-\nu),2)$ such that

  $$0< \bigl(\nabla W(t, x), x\bigr)\le\mu W(t, x), \quad \forall (t, x)\in { \mathbb{R}}\times{\mathbb{R}}^{N} \setminus\{0\}; $$

• (A4) There exist three constants $a_{1}, r_{1}>0$ and $1<\mu_{1}\in (2/(3-\nu), \mu]$ such that

  $$W(t, x)\ge a_{1}|x|^{\mu_{1}}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}, |x|\ge r_{1}; $$

• (A5) $W(t, 0)\equiv0$ and there exist three constants $a_{2}, r_{2}>0$ and $1<\mu_{2}\in(2/(3-\nu), \mu]$ such that

  $$\bigl|\nabla W(t, x)\bigr|\le a_{2}|x|^{\mu_{2}-1}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}, |x|\le r_{2}. $$

Then system (1.1) has at least one nontrivial homoclinic solution. Moreover, if $W(t, x)$ is also even with respect to x, then system (1.1) has infinitely many homoclinic solutions.

In Theorem A, assumptions (A2)-(A5) imply that there exist positive constants $a_{*}$, $a^{*}$, $b_{*}$ and $b^{*}$ such that

$$\begin{aligned}& b_{*}|x|^{\mu}\le W(t, x)\le a_{*}|x|^{\mu_{2}}, \quad \forall (t, x) \in \mathbb {R}\times{ \mathbb {R}}^{N}, |x|\le1, \end{aligned}$$

(1.2)

$$\begin{aligned}& a^{*}|x|^{\mu_{1}}\le W(t, x)\le b^{*}|x|^{\mu}, \quad \forall (t, x) \in \mathbb {R}\times{ \mathbb {R}}^{N}, |x|\ge1. \end{aligned}$$

(1.3)

However, there are many potential functions $W(t, x)$ satisfying (1.2) and (1.3), but not (A3). For example $W(t, x)= (1+\sin^{2}t) (2|x|^{5/4} -3 |x|^{3/2} + 2|x|^{7/4} )$ is such a potential function. In particular, Theorem A is only applicable when the potential $W(t,x)$ is positive definite.

In the present paper, we will use new tricks to generalize and improve Theorem A. For example, we can replace (A1) by a weaker one ($\mathrm{L}_{\nu}$):

($\mathrm{L}_{\nu}$):

There exists a constant $\nu<2$ such that

$$\liminf_{|t|\to+\infty} \Bigl[|t|^{\nu-2}\inf _{|x|=1}\bigl(L(t)x, x\bigr) \Bigr] >0. $$

We also relax (A3) and (A4) in Theorem A to two of the following weaker assumptions:

(W3):

There exist constants $b_{1}, b_{2}>0$ and $\max\{1, 2/(3-\nu )\} <\gamma_{5}\le\gamma_{4} < 2$ such that

$$2W(t, x)-\bigl(\nabla W(t, x), x\bigr)\ge \left \{ \textstyle\begin{array}{@{}l@{\quad}l} b_{1}|x|^{\gamma_{4}}, & |x|\le1,\\ b_{2}|x|^{\gamma_{5}}, & |x|\ge1, \end{array}\displaystyle \right . \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}; $$

(W4):

There exist constants $b_{3}, b_{4}>0$ and $\max\{1, 2/(3-\nu )\} <\gamma_{7}\le\gamma_{6} < 2$ such that

$$W(t, x)\ge \left \{ \textstyle\begin{array}{@{}l@{\quad}l} b_{3}|x|^{\gamma_{6}}, & |x|\le1,\\ b_{4}|x|^{\gamma_{7}}, & |x|\ge1, \end{array}\displaystyle \right . \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}; $$

(W3′):

There exist constants $b_{5}>0$ , $b_{6}\ge0$ and $\max\{ 1, 2/(3-\nu)\} <\gamma_{9} < \gamma_{8} < 2$ such that

$$2W(t, x)-\bigl(\nabla W(t, x), x\bigr)\ge b_{5}|x|^{\gamma_{8}}- b_{6}|x|^{\gamma_{9}}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}; $$

(W4′):

There exist constants $b_{7}>0$ , $b_{8}\ge0$ and $\max\{ 1, 2/(3-\nu)\} <\gamma_{10} < \gamma_{11}< 2$

$$W(t, x)\ge b_{7}|x|^{\gamma_{10}}- b_{8}|x|^{\gamma_{11}}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}; $$

(W4″):

$\lim_{|x|\to 0}\frac{W(t, x)}{|x|^{2}}=\infty$ uniformly in $t\in \mathbb {R}$.

Our main results are the following four theorems.

Theorem 1.1

Assume that L and W satisfy ($\mathrm{L}_{\nu }$), (W3), (W4) and the following conditions:

• (W1) There exist constants $\max\{1, 2/(3-\nu)\} <\gamma _{1}<\gamma_{2} < 2$ and $a_{1}, a_{2} \ge0$ such that

  $$\bigl|W(t, x)\bigr|\le a_{1}|x|^{\gamma_{1}}+a_{2}|x|^{\gamma_{2}}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}; $$

• (W2) There exists a function $\varphi\in C([0, +\infty), [0, +\infty))$ such that

  $$\bigl|\nabla W(t, x)\bigr|\le\varphi\bigl(|x|\bigr), \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}, $$

  where $\varphi(s)=O(s^{\gamma_{3}-1})$ as $s\to0^{+}$, $\max\{ 1, 2/(3-\nu)\} <\gamma_{3} < 2$.

Then system (1.1) possesses at least one nontrivial homoclinic solution.

Theorem 1.2

Assume that L and W satisfy ($\mathrm{L}_{\nu }$), (W1), (W2), (W3), (W4) and the following condition:

• (W5) $W(t, -x)= W(t, x)$, $\forall (t, x)\in{\mathbb {R}}\times{\mathbb{R}}^{N}$.

Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.

Theorem 1.3

Assume that L and W satisfy ($\mathrm{L}_{\nu }$), (W1), (W2), (W3′) and (W4). Then system (1.1) possesses at least one nontrivial homoclinic solution.

Theorem 1.4

Assume that L and W satisfy ($\mathrm{L}_{\nu }$), (W1), (W2), (W3′), (W4′) and (W5). Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.

Corollary 1.5

The conclusion of Theorem 1.4 also holds if (W4′) is replaced by (W4″).

Remark 1.6

Our results can be applied to the following potential functions:

$$ W(t, x)= \bigl(1+\sin^{2}t\bigr) \bigl(|x|^{5/4} -3 |x|^{3/2} + |x|^{7/4} \bigr) $$

(1.4)

and

$$ W(t, x)=d_{1}|x|^{\tau_{1}} -\sum_{i=2}^{m-1}d_{i}|x|^{\tau _{i}}+d_{m}|x|^{\tau_{m}}, $$

(1.5)

where $m\ge4$, $1<\tau_{1}<\tau_{2} < \cdots< \tau_{m}<2$ and $d_{i}>0$ for $i=1, 2, \ldots, m$. Note that the above potential functions are with indefinite signs, and hence Theorem A is not applicable. See Examples 4.1 and 4.2 in Section 4.

The remainder of this paper is organized as follows. In Section 2, we first define a Hilbert space E and describe its space structure. Then we state the critical point theorems needed for the proofs of our main results. The proofs of our main results are given in Section 3. Some examples to illustrate our results are given in Section 4.

Throughout this paper, we denote the norm of $L^{p}(\mathbb {R}, \mathbb {R}^{N})$ by $\|u\| _{p}=(\int_{\mathbb {R}}|u|^{s}\,\mathrm{d}t)^{1/p}$ for $p\ge1$, and positive constants, possibly different in different places, by $C_{1}, C_{2}, \ldots$ .

2 Preliminaries

In this section, we first make the following weaker assumption on $L(t)$:

• (L) The smallest eigenvalue of $L(t) \to+\infty$ as $|t| \to +\infty$ , i.e.,

  $$\lim_{|t|\to+\infty} \Bigl[\inf_{|x|=1}\bigl(L(t)x, x \bigr) \Bigr] = + \infty. $$

In order to establish our existence results via the critical point theory, we first describe some properties of the space on which the variational functional associated with (1.1) is defined.

In what follows $L(t)$ is assumed to satisfy assumption (L). We denote by $I_{N}$ the identity matrix of order N, I the identity operator. Let $\{\mathcal{E}(\lambda): -\infty< \lambda< +\infty\}$ and $|\mathcal{A}|$ be the spectral family and the absolute value of $\mathcal{A}$, respectively, and $|\mathcal{A}|^{1/2}$ be the square root of $|\mathcal{A}|$. Set $U=I-\mathcal{E}(0)-\mathcal{E}(0-)$. Then U commutes with $\mathcal{A}$, $|\mathcal{A}|$ and $|\mathcal{A}|^{1/2}$, and $\mathcal{A} = U|\mathcal{A}|$ is the polar decomposition of $\mathcal{A}$ (see [22, 23]). Let $E =\mathfrak{D}(|\mathcal{A}|^{1/2})$, the domain of $|\mathcal{A}|^{1/2}$, and define on E the inner product

$$(u, v)_{0}= \bigl(|\mathcal{A}|^{1/2}u, | \mathcal{A}|^{1/2}v \bigr)_{2}+(u, v)_{2}, \quad \forall u, v\in E $$

and the norm

$$\|u\|_{0}=\sqrt{(u, u)_{0}}, \quad \forall u\in E, $$

where, as usual, $(\cdot, \cdot)_{2}$ denotes the inner product of $L^{2}$. Then E is a Hilbert space. Clearly, $C_{0}^{\infty}\equiv C_{0}^{\infty}(\mathbb {R}, {\mathbb {R}}^{N})$ is dense in E.

By (L), $L(t)$ is bounded from below and so there is $l_{0}>0$ such that

$$ l(t)+l_{0}\ge1, \quad \forall t\in \mathbb {R}, $$

(2.1)

where, and in the sequel,

$$ l(t)=\inf_{x\in{ \mathbb {R}}^{N}, |x|=1}\bigl(L(t)x, x\bigr). $$

(2.2)

Set

$$\begin{aligned}& E_{*}= \biggl\{ u\in W^{1,2}\bigl(\mathbb {R}, {\mathbb {R}}^{N}\bigr) : \int _{\mathbb{R}} \bigl[|\dot {u}|^{2}+\bigl( \bigl(L(s)+l_{0}I_{N}\bigr)u, u\bigr) \bigr]\,\mathrm{d}s< + \infty \biggr\} , \\& (u, v)_{*}= \int_{\mathbb{R}} \bigl[(\dot{u}, \dot{v})+\bigl( \bigl(L(s)+l_{0}I_{N}\bigr)u, v\bigr) \bigr]\,\mathrm{d}s, \quad \forall u, v\in E_{*} \end{aligned}$$

and

$$\|u\|_{*}= \biggl\{ \int_{\mathbb{R}} \bigl[|\dot{u}|^{2}+ \bigl(\bigl(L(s)+l_{0}I_{N}\bigr)u, u\bigr) \bigr]\, \mathrm{d}s \biggr\} ^{1/2},\quad \forall u\in E_{*}. $$

Then $E_{*}$ is also a Hilbert space with the above inner product $(\cdot , \cdot)_{*}$ and the norm $\|\cdot\|_{*}$.

Lemma 2.1

([15])

For $u\in E_{*}$,

$$\begin{aligned}& \|u\|_{\infty} \le\frac{1}{\sqrt{2}}\|u\|_{*}=\frac{1}{\sqrt{2}} \biggl\{ \int _{\mathbb{R}} \bigl[|\dot{u}|^{2}+\bigl( \bigl(L(s)+l_{0}I_{N}\bigr)u, u\bigr) \bigr]\,\mathrm{d}s \biggr\} ^{1/2}, \end{aligned}$$

(2.3)

$$\begin{aligned}& \bigl|u(t)\bigr| \le \biggl\{ \int_{t}^{\infty}\frac{1}{\sqrt{l(s)+l_{0}}} \bigl[|\dot {u}|^{2}+\bigl(\bigl(L(s)+l_{0}I_{N} \bigr)u, u\bigr) \bigr]\,\mathrm{d}s \biggr\} ^{1/2},\quad \forall t\in{ \mathbb{R}} \end{aligned}$$

(2.4)

and

$$ \bigl|u(t)\bigr| \le \biggl\{ \int_{-\infty}^{t} \frac{1}{\sqrt{l(s)+l_{0}}} \bigl[|\dot{u}|^{2}+\bigl(\bigl(L(s)+l_{0}I_{N} \bigr)u, u\bigr) \bigr]\,\mathrm{d}s \biggr\} ^{1/2},\quad \forall t\in{ \mathbb{R}}. $$

(2.5)

Lemma 2.2

Suppose that $L(t)$ satisfies (L). Then E is compactly embedded in $L^{p}(\mathbb {R}, {\mathbb {R}}^{N})$ for $2\le p\le\infty$, and

$$\|u\|_{p}^{p}\le2^{(2-p)/2}\|u \|_{*}^{p}, \qquad \int_{|t|>T}\bigl|u(t)\bigr|^{p}\, \mathrm{d}t\le\frac{2^{(2-p)/2}}{\min_{|s|\ge T}[l(s)+l_{0}]}\|u\|_{*}^{p}, \quad \forall T > 0. $$

(2.6)

Proof

In fact, the first part of Lemma 2.2 was proved in [21]. Here, we give the proof of the second part. From (2.1), (2.2) and (2.3), we have

$$ \|u\|_{p}^{p} \le\|u\|_{\infty}^{p-2} \int_{\mathbb {R}}\bigl|u(t)\bigr|^{2}\,\mathrm{d}t \le\|u \|_{\infty}^{p-2}\int_{\mathbb {R}}\bigl( \bigl(L(t)+l_{0}I_{N}\bigr)u, u\bigr)\,\mathrm{d}t \le 2^{(2-p)/2}\|u\|_{*}^{p} $$

(2.7)

and

$$\begin{aligned} \int_{|t|>T}\bigl|u(t)\bigr|^{p}\,\mathrm{d}t \le& \|u \|_{\infty}^{p-2}\int_{|t|>T}\bigl|u(t)\bigr|^{2} \,\mathrm{d}t \\ \le& \|u\|_{\infty}^{p-2}\int_{|t|>T} \frac{((L(t)+l_{0}I_{N})u, u)}{l(t)+l_{0}}\,\mathrm{d}t \\ \le& \frac{\|u\|_{\infty}^{p-2}}{\min_{|s|\ge T}[l(s)+l_{0}]}\|u\|_{*}^{2} \le\frac{2^{(2-p)/2}}{\min_{|s|\ge T}[l(s)+l_{0}]}\|u \|_{*}^{p}. \end{aligned}$$

 □

By (L), there exists a constant $\alpha\in \mathbb {R}$ such that

$$ \bigl(L(t)x, x\bigr) > \alpha|x|^{2}, \quad \forall x \in{ \mathbb {R}}^{N}\setminus\{ 0\}. $$

(2.8)

Analogous to the proof of [24], Lemma 2.4, we can prove the following lemma by using Lemma 2.2.

Lemma 2.3

Suppose that $L(t)$ satisfies (L). Let

$$ E^{-}=\mathcal{E}(0-)E, \qquad E^{0}=\bigl[ \mathcal{E}(0)-\mathcal {E}(0-)\bigr]E, \qquad E^{+}=\bigl[ \mathcal{E}(+\infty)-\mathcal{E}(0)\bigr]E. $$

(2.9)

Then $E=E^{-}\oplus E^{0}\oplus E^{+}$, and $E^{-}$, $E^{0}$ and $E^{+}$ are orthogonal with respect to the inner products $(\cdot, \cdot)_{0}$ and $(\cdot, \cdot)_{2}$ on E. Furthermore, the following hold:

$$\begin{aligned}& \dim\bigl(\mathcal{E}(M)E\bigr)< +\infty, \quad \forall M\ge0, \end{aligned}$$

(2.10)

$$\begin{aligned}& E^{0}=\operatorname{Ker}(\mathcal{A}), \qquad \mathcal{A}u^{-}=-| \mathcal{A}|u^{-}, \qquad \mathcal{A}u^{+}=| \mathcal{A}|u^{+}, \quad \forall u\in \mathfrak{D}( \mathcal{A}) \end{aligned}$$

(2.11)

and

$$ u=u^{-}+u^{0}+u^{+}, \quad \forall u \in E, $$

(2.12)

where

$$ \begin{aligned} &u^{-}=\mathcal{E}(0-)u\in E^{-}, \qquad u^{0}=\bigl[\mathcal{E}(0)-\mathcal {E}(0-)\bigr]u\in E^{0},\\ &u^{+}=\bigl[\mathcal{E}(+\infty)- \mathcal{E}(0)\bigr]u\in E^{+}. \end{aligned} $$

(2.13)

In view of Lemma 2.3, we introduce on E the following inner product:

$$(u, v)= \bigl(|\mathcal{A}|^{1/2}u, |\mathcal{A}|^{1/2}v \bigr)_{2}+\bigl(u^{0}, v^{0} \bigr)_{2} $$

and the norm

$$\|u\|^{2}= (u, u)=\bigl\Vert |\mathcal{A}|^{1/2}u\bigr\Vert _{2}^{2}+\bigl\| u^{0}\bigr\| _{2}^{2}, $$

where $u=u^{-}+u^{0}+u^{+}$, $v=v^{-}+v^{0}+v^{+} \in E^{-}\oplus E^{0}\oplus E^{+}=E$. Then it is easy to check the following lemma.

Lemma 2.4

Suppose that $L(t)$ satisfies (L). Then $E^{-}$, $E^{0}$ and $E^{+}$ are orthogonal with respect to the inner product $(\cdot, \cdot)$ on E.

Analogous to the proof of [24], Lemma 2.1, Lemma 2.6, we can prove the following lemma.

Lemma 2.5

Suppose that $L(t)$ satisfies (L). Then the norms $\|\cdot\|_{0}$, $\|\cdot\|_{*}$ and $\|\cdot\|$ on E are equivalent. Hence, there exists $\beta>0$ such that

$$ \|u\|_{*}\le\beta\|u\|, \quad \forall u\in E. $$

(2.14)

By virtue of ($\mathrm{L}_{\nu}$), there exist two constants $T_{0}>0$ and $M_{0}>0$ such that

$$ |t|^{\nu-2}l(t)=|t|^{\nu-2}\inf_{|x|=1}\bigl(L(t)x, x\bigr) \ge M_{0}, \quad \forall |t|\ge T_{0}, $$

which implies

$$ |t|^{\nu-2}\bigl(L(t)x, x\bigr) \ge M_{0}|x|^{2}, \quad \forall |t|\ge T_{0}, x\in{ \mathbb {R}}^{N}. $$

(2.15)

Lemma 2.6

Suppose that $L(t)$ satisfies ($\mathrm{L}_{\nu}$). Then, for $1\le p\in(2/(3-\nu), 2)$, E is compactly embedded in $L^{p}(\mathbb {R}, \mathbb {R}^{N})$; moreover,

$$ \int_{|t|\ge T}\bigl|u(t)\bigr|^{p}\,\mathrm{d}t \le\frac{K(p)}{T^{\kappa}}\|u\| _{*}^{p}, \quad \forall u\in E, T\ge T_{0} $$

(2.16)

and

$$ \|u\|_{p}^{p} \le \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-p/(2-p)}\, \mathrm{d}t \biggr)^{1-\frac{p}{2}}+\frac{K(p)}{T^{\kappa}} \biggr]\|u \|_{*}^{p}, \quad \forall u\in E, T\ge T_{0}, $$

(2.17)

where

$$ \kappa=\frac{(3-\nu)p-2}{2}>0, \qquad K(p)= \biggl[\frac {2(2-p)}{(3-\nu )p-2} \biggr]^{1-\frac{p}{2}}M_{0}^{-p/2}. $$

(2.18)

Proof

For $1\le p\in(2/(3-\nu), 2)$, we set $r=[(3-\nu )p-2]/(2-p)$. Then $r>0$. For $u\in E$ and $T\ge T_{0}$, it follows from (2.15) and the Hölder inequality that

$$\begin{aligned} \int_{|t|\ge T}\bigl|u(t)\bigr|^{p}\,\mathrm{d}t \le& \biggl( \int_{|t|\ge T}|t|^{-(2-\nu)p/(2-p)}\,\mathrm{d}t \biggr)^{1-\frac{p}{2}} \biggl(\int_{|t|\ge T}|t|^{2-\nu}\bigl|u(t)\bigr|^{2} \,\mathrm{d}t \biggr)^{\frac {p}{2}} \\ \le& \biggl(\frac{2}{rT^{r}} \biggr)^{1-\frac{p}{2}} \biggl[ \frac{1}{M_{0}}\int_{|t|\ge T}\bigl(L(t)u(t), u(t)\bigr)\, \mathrm{d}t \biggr]^{\frac{p}{2}} \\ \le& \frac{2^{(2-p)/2}}{M_{0}^{p/2}r^{(2-p)/2}T^{\kappa}}\|u\|_{*}^{p}= \frac{K(p)}{T^{\kappa}}\|u \|_{*}^{p}. \end{aligned}$$

From (2.2) and (2.16), one has

$$\begin{aligned} \|u\|_{p}^{p} = & \int_{|t|\le T}\bigl|u(t)\bigr|^{p} \,\mathrm{d}t+\int_{|t|> T}\bigl|u(t)\bigr|^{p}\,\mathrm{d}t \\ \le& \biggl(\int_{|t|\le T}\bigl[l(t)+l_{0} \bigr]^{-p/(2-p)}\,\mathrm{d}t \biggr)^{1-\frac{p}{2}} \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]\bigl|u(t)\bigr|^{2}\, \mathrm{d}t \biggr)^{\frac {p}{2}}+\frac{K(p)}{T^{\kappa}}\|u\|_{*}^{p} \\ \le& \biggl(\int_{|t|\le T}\bigl[l(t)+l_{0} \bigr]^{-p/(2-p)}\,\mathrm{d}t \biggr)^{1-\frac{p}{2}}\|u\|_{*}^{p}+ \frac{K(p)}{T^{\kappa}}\|u\|_{*}^{p}. \end{aligned}$$

For $1\le p\in(2/(3-\nu), 2)$, applying (2.16), we can prove that E is compactly embedded in $L^{p}(\mathbb {R}, \mathbb {R}^{N})$ by a standard argument. □

Lemma 2.7

([25])

Let X be real Banach space, Q and S be two closed subsets of X, and S and ∂Q link. Suppose that $f\in C^{1}(X, {\mathbb {R}})$ satisfy the (PS)-condition, and that

• (i) there exist two constants $\eta>\zeta$ such that $\sup_{x\in\partial Q}f(x)\le\zeta< \eta\le\inf_{x\in S}f(x)$;

• (ii) $\sup_{x\in Q}f(x)<+\infty$.

Then f possesses a critical value $c\ge\eta$.

Lemma 2.8

([21], Lemma 2.4)

Let X be an infinite dimensional Banach space and $f\in C^{1}(X, {\mathbb {R}})$ be even, satisfy the (PS)-condition, and $f(0)=0$. If $X=X_{1}\oplus X_{2}$, where $X_{1}$ is finite dimensional, and f satisfies

• (i) f is bounded from below on $X_{2}$;

• (ii) for each finite dimensional subspace $\tilde{X}\subset X$, there are positive constants $\rho=\rho(\tilde{X})$ and $\sigma=\sigma(\tilde{X})$ such that $f|_{B_{\rho}\cap\tilde {X}}\le0$ and $f|_{\partial B_{\rho}\cap\tilde{X}} \le-\sigma$, where $B_{\rho}=\{x\in X : \|x\|=\rho\}$.

Then f possesses infinitely many nontrivial critical points.

Lemma 2.9

([26])

Let X be a real Banach space and $f\in C^{1}(X, {\mathbb {R}})$ satisfy the (PS)-condition. If f is bounded from below, then $c=\inf_{X}f$ is a critical value of f.

3 Proofs of theorems

Lemma 3.1

Assume that ($\mathrm{L}_{\nu}$) and (W1) hold. Then, for $u\in E$,

$$ \int_{\mathbb {R}}\bigl|W(t, u)\bigr|\,\mathrm{d}t\le \phi_{1}(T)\|u\|^{\gamma_{1}}+\phi _{2}(T)\| u \|^{\gamma_{2}}, \quad T\ge T_{0}, $$

(3.1)

where

$$\begin{aligned}& \phi_{1}(T)=a_{1}\beta^{\gamma_{1}} \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-\gamma_{1}/(2-\gamma_{1})}\, \mathrm{d}t \biggr)^{1-\frac {\gamma_{1}}{2}} +\frac{K(\gamma_{1})}{T^{\kappa_{1}}} \biggr], \end{aligned}$$

(3.2)

$$\begin{aligned}& \phi_{2}(T)=a_{2}\beta^{\gamma_{2}} \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-\gamma_{2}/(2-\gamma_{2})}\, \mathrm{d}t \biggr)^{1-\frac {\gamma_{2}}{2}} +\frac{K(\gamma_{2})}{T^{\kappa_{2}}} \biggr] \end{aligned}$$

(3.3)

and

$$ \kappa_{1}=\frac{(3-\nu)\gamma_{1}-2}{2}, \qquad \kappa_{2}=\frac {(3-\nu )\gamma_{2}-2}{2}. $$

(3.4)

Proof

For $T\ge T_{0}$, it follows from (2.14), (2.17), (3.2), (3.3) and (W1) that

$$\begin{aligned} \int_{\mathbb {R}}\bigl|W(t, u)\bigr|\,\mathrm{d}t \le& a_{1}\int _{\mathbb {R}}\bigl|u(t)\bigr|^{\gamma_{1}}\,\mathrm{d}t+a_{2}\int _{\mathbb {R}}\bigl|u(t)\bigr|^{\gamma_{2}}\,\mathrm{d}t \\ \le& a_{1} \biggl[ \biggl(\int_{|t|\le T} \bigl[l(t)+l_{0}\bigr]^{-\gamma _{1}/(2-\gamma _{1})}\,\mathrm{d}t \biggr)^{1-\frac{\gamma_{1}}{2}} +\frac{K(\gamma_{1})}{T^{\kappa_{1}}} \biggr]\|u\|_{*}^{\gamma _{1}} \\ &{} +a_{2} \biggl[ \biggl(\int_{|t|\le T} \bigl[l(t)+l_{0}\bigr]^{-\gamma_{2}/(2-\gamma _{2})}\,\mathrm{d}t \biggr)^{1-\frac{\gamma_{2}}{2}} +\frac{K(\gamma_{2})}{T^{\kappa_{2}}} \biggr]\|u\|_{*}^{\gamma _{2}} \\ \le& a_{1}\beta^{\gamma_{1}} \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-\gamma_{1}/(2-\gamma_{1})}\, \mathrm{d}t \biggr)^{1-\frac {\gamma_{1}}{2}} +\frac{K(\gamma_{1})}{T^{\kappa_{1}}} \biggr]\|u\|^{\gamma_{1}} \\ &{} +a_{2}\beta^{\gamma_{2}} \biggl[ \biggl(\int _{|t|\le T}\bigl[l(t)+l_{0}\bigr]^{-\gamma _{2}/(2-\gamma_{2})}\, \mathrm{d}t \biggr)^{1-\frac{\gamma_{2}}{2}} +\frac{K(\gamma_{2})}{T^{\kappa_{2}}} \biggr]\|u\|^{\gamma_{2}} \\ = & \phi_{1}(T)\|u\|^{\gamma_{1}}+\phi_{2}(T)\|u \|^{\gamma_{2}}. \end{aligned}$$

 □

Analogous to the proof of [17], Lemma 2.2, we can prove the following lemma.

Lemma 3.2

Assume that ($\mathrm{L}_{\nu}$), (W1) and (W2) hold. Then the functional $f: E\rightarrow{ \mathbb {R}}$ defined by

$$ \Phi(u)=\frac{1}{2} \bigl(\bigl\| u^{+} \bigr\| ^{2}-\bigl\| u^{-}\bigr\| ^{2} \bigr)-\int _{\mathbb {R}}W(t, u)\,\mathrm{d}t $$

(3.5)

is well defined and of class $C^{1}(E, {\mathbb {R}})$ and

$$ \bigl\langle \Phi'(u), v \bigr\rangle = \bigl\| u^{+}\bigr\| ^{2}-\bigl\| u^{-}\bigr\| ^{2} -\int _{\mathbb {R}}\bigl(\nabla W(t, u), v\bigr)\,\mathrm{d}t. $$

(3.6)

Furthermore, the critical points of Φ in E are classical solutions of (1.1) with $u(\pm\infty)=0$.

Proof of Theorem 1.1

In view of Lemma 3.2, $\Phi\in C^{1}(E, \mathbb {R})$. In what follows, we divide the rest of the proof of Theorem 1.1 into four steps.

Step 1. Φ satisfies the (PS)-condition.

Assume that $\{u_{n}\}_{n\in{\mathbb{N}}}\subset E$ is a (PS)-sequence: $\{\Phi(u_{n})\}_{k\in{\mathbb{N}}}$ is bounded and $\|\Phi'(u_{n})\| \rightarrow0$ as $n\rightarrow+\infty$. In the sequel we write for any $u\in E$

$$ u^{1}(t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(t) & \mbox{if } |u(t)|< 1,\\ 0 & \mbox{if } |u(t)| \ge1; \end{array}\displaystyle \displaystyle \right . \qquad u^{2}(t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0 & \mbox{if } |u(t)|< 1,\\ u(t) & \mbox{if } |u(t)| \ge1. \end{array}\displaystyle \displaystyle \displaystyle \displaystyle \displaystyle \right . $$

(3.7)

Then, by (3.5), (3.6), (3.7) and (W3), we get

$$\begin{aligned} \bigl\langle \Phi'(u_{n}), u_{n} \bigr\rangle -2\Phi(u_{n}) = & \int_{\mathbb {R}}\bigl[2W(t, u_{n})-\bigl(\nabla W(t, u_{n}), u_{n}\bigr) \bigr]\,\mathrm{d}t \\ \ge& b_{1}\int_{\mathbb {R}}\bigl|u^{1}_{n}\bigr|^{\gamma_{4}} \,\mathrm{d}t+b_{2}\int_{\mathbb {R}}\bigl|u^{2}_{n}\bigr|^{\gamma_{5}} \,\mathrm{d}t \\ = & b_{1}\bigl\| u^{1}_{n}\bigr\| _{\gamma_{4}}^{\gamma_{4}}+b_{2} \bigl\| u^{2}_{n}\bigr\| _{\gamma _{5}}^{\gamma_{5}}. \end{aligned}$$

It follows that there exists a constant $C_{1}>0$ such that

$$ b_{1}\bigl\| u^{1}_{n} \bigr\| _{\gamma_{4}}^{\gamma_{4}}+b_{2}\bigl\| u^{2}_{n} \bigr\| _{\gamma _{5}}^{\gamma _{5}}\le C_{1}\bigl(1+\|u_{n}\|\bigr). $$

(3.8)

Since $\dim(E^{-}\oplus E^{0})<+\infty$, there exists a constant $C_{2}>0$ such that

$$\begin{aligned} \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}^{2} = & \bigl(u_{n}^{-}+u_{n}^{0}, u_{n} \bigr)_{2} \\ = & \bigl(u_{n}^{-}+u_{n}^{0}, u^{1}_{n} \bigr)_{2}+ \bigl(u_{n}^{-}+u_{n}^{0}, u^{2}_{n} \bigr)_{2} \\ \le& \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{\gamma_{4}'}\bigl\| u^{1}_{n}\bigr\| _{\gamma_{4}}+\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{\gamma_{5}'}\bigl\| u^{2}_{n}\bigr\| _{\gamma_{5}} \\ \le& C_{2}\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2} \bigl(\bigl\| u^{1}_{n}\bigr\| _{\gamma_{4}}+\bigl\| u^{2}_{n}\bigr\| _{\gamma_{5}} \bigr), \end{aligned}$$

(3.9)

where $\gamma_{4}'=\gamma_{4}/(\gamma_{4}-1)$ and $\gamma_{5}'=\gamma _{5}/(\gamma _{5}-1)$. Combining (3.8) with (3.9), one has

$$ \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}\le C_{3}\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}^{2}\le C_{4} \bigl(1+\| u_{n} \|^{2/\gamma_{4}}+\|u_{n}\|^{2/\gamma_{5}} \bigr). $$

(3.10)

Choose $T_{2}>T_{0}$, it follows from (3.1) that

$$ \int_{\mathbb {R}}W(t, u_{n})\,\mathrm{d}t \le\phi_{1}(T_{2})\|u_{n}\|^{\gamma _{1}}+\phi _{2}(T_{2})\|u_{n}\|^{\gamma_{2}}. $$

(3.11)

From (3.5), (3.10) and (3.11), we obtain

$$\begin{aligned} \|u_{n}\|^{2} = & \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}+\bigl\| u_{n}^{+}\bigr\| ^{2} \\ = & \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}+2\Phi(u_{n})+\bigl\| u_{n}^{-} \bigr\| ^{2}+2\int_{\mathbb {R}}W(t, u_{n})\, \mathrm{d}t \\ \le& 2C_{4} \bigl(1+\|u_{n}\|^{2/\gamma_{4}}+ \|u_{n}\|^{2/\gamma_{5}} \bigr)+2\Phi(u_{n}) \\ &{} +2\phi_{1}(T_{2})\|u_{n}\|^{\gamma_{1}}+2 \phi_{2}(T_{2})\|u_{n}\|^{\gamma _{2}} \\ \le& C_{5} \bigl(1+\|u_{n}\|^{\gamma_{1}}+ \|u_{n}\|^{\gamma_{2}}+\|u_{n}\| ^{2/\gamma_{4}}+ \|u_{n}\|^{2/\gamma_{5}} \bigr). \end{aligned}$$

(3.12)

Since $1<\gamma_{1}<\gamma_{2}<2$, $1<\gamma_{5} \le\gamma_{4}<2$, it follows from (3.12) that $\{\|u_{n}\|\}$ is bounded, and so $\{\|u_{n}\|_{*}\}$ is bounded. Choose a constant $\Lambda>0$ such that

$$ \|u_{n}\|_{\infty}\le\frac{1}{\sqrt{2}} \|u_{n}\|_{*}\le\Lambda, \quad n\in\mathbb{N}. $$

(3.13)

Passing to a subsequence if necessary, it can be assumed that $u_{n}\rightharpoonup u_{0}$ in E. Hence $u_{n}\rightarrow u_{0}$ in $L^{\infty}_{\mathrm {\mathrm{loc}}}(\mathbb {R}, {\mathbb {R}}^{N})$; moreover, it is easy to verify that $\{u_{n}(t)\}$ converges to $u_{0}(t)$ point-wise for all $t\in{\mathbb{R}}$. Hence, (3.13) yields that $\|u_{0}\|_{\infty}\le\Lambda$. By (W2), there exists $M_{3}>0$ such that

$$ \nabla W(t, x)\le M_{3}|x|^{\gamma_{3}-1}, \quad \forall x\in{ \mathbb {R}}^{N}, |x|\le\Lambda. $$

(3.14)

For any given number $\varepsilon>0$, we can choose $T_{3}>T_{0}$ such that

$$ \frac{K(\gamma_{3}) [ (\sqrt{2}\Lambda )^{\gamma _{3}}+\|u_{0}\| _{*}^{\gamma_{3}} ]}{T_{3}^{\kappa_{3}}}< \varepsilon. $$

(3.15)

Hence, from (2.16), (3.13), (3.14) and (3.15) we have that

$$\begin{aligned} \int_{|t|>T_{3}}\bigl|\nabla W(t, u_{n})- \nabla W(t, u_{0})\bigr||u_{n}-u_{0}|\,\mathrm{d}t \le& 2M_{3}\int_{|t|>T_{3}}\bigl(\bigl|u_{k}(t)\bigr|^{\gamma_{3}}+\bigl|u_{0}(t)\bigr|^{\gamma _{3}} \bigr)dt \\ \le& \frac{2M_{3}K(\gamma_{3})}{T_{3}^{\kappa_{3}}} \bigl(\|u_{k}\| _{*}^{\gamma _{3}}+ \|u_{0}\|_{*}^{\gamma_{3}} \bigr) \\ \le& \frac{2M_{3}K(\gamma_{3})}{T_{3}^{\kappa_{3}}} \bigl[ (\sqrt {2}\Lambda )^{\gamma_{3}}+ \|u_{0}\|_{*}^{\gamma_{3}} \bigr] \\ \le& 2M_{3}\varepsilon, \quad n\in{\mathbb{N}}. \end{aligned}$$

(3.16)

On the other hand, since $u_{n}\rightarrow u_{0}$ in $L^{\infty}_{\mathrm {\mathrm{loc}}}(\mathbb {R}, {\mathbb {R}}^{N})$, it follows from the continuity of $\nabla W(t, x)$ that

$$ \int_{-T_{3}}^{T_{3}}\bigl|\nabla W(t, u_{n})-\nabla W(t, u_{0})\bigr||u_{n}-u_{0}| \,\mathrm{d}t = o(1). $$

(3.17)

Since ε is arbitrary, combining (3.16) with (3.17) we get

$$ \int_{\mathbb{R}}\bigl(\nabla W(t, u_{n})-\nabla W(t, u_{0}), u_{n}-u_{0} \bigr)\,\mathrm{d}t = o(1). $$

(3.18)

It follows from (3.6) that

$$\begin{aligned} \bigl\langle \Phi'(u_{n})- \Phi'(u_{0}), u_{n}-u_{0}\bigr\rangle = & \bigl\| u_{n}^{+}-u_{0}^{+} \bigr\| ^{2}-\bigl\| u_{n}^{-}-u_{0}^{-} \bigr\| ^{2} \\ &{} -\int_{{\mathbb{R}}}\bigl(\nabla W(t, u_{n})-\nabla W(t, u_{0}), u_{n}-u_{0}\bigr)\,\mathrm{d}t. \end{aligned}$$

(3.19)

Since $\langle\Phi'(u_{n})-\Phi'(u_{0}), u_{n}-u_{0}\rangle=o(1)$, it follows from (3.18) and (3.19) that

$$ \bigl\| u_{n}^{+}-u_{0}^{+} \bigr\| ^{2}-\bigl\| u_{n}^{-}-u_{0}^{-} \bigr\| ^{2}=o(1). $$

(3.20)

Since $u_{n}\rightharpoonup u_{0}$ in E and $\dim(E^{-}\oplus E^{0})<+\infty$, it follows that

$$ \bigl\| u_{n}^{0}-u_{0}^{0} \bigr\| ^{2}+\bigl\| u_{n}^{-}-u_{0}^{-} \bigr\| ^{2}= o(1). $$

(3.21)

Combining (3.20) with (3.21), we have

$$ \|u_{n}-u_{0}\|^{2}=\bigl\| u_{n}^{+}-u_{0}^{+} \bigr\| ^{2}+\bigl\| u_{n}^{0}-u_{0}^{0} \bigr\| ^{2}+\bigl\| u_{n}^{-}-u_{0}^{-} \bigr\| ^{2}=o(1). $$

Hence, Φ satisfies the (PS)-condition.

Step 2. $\Phi(u)\rightarrow+\infty$ as $\|u\|\rightarrow+\infty$ and $u\in E^{+}$.

It follows from (3.1) that

$$ \int_{\mathbb {R}}W(t, u)\,\mathrm{d}t \le \phi_{1}(T_{2})\|u\|^{\gamma_{1}}+\phi _{2}(T_{2}) \|u\|^{\gamma_{2}}, \quad \forall u\in E. $$

(3.22)

Hence, for $u\in E^{+}$, it follows from (3.5) and (3.22) that

$$\begin{aligned} \Phi(u) = & \frac{1}{2}\|u\|^{2}-\int_{\mathbb {R}}W(t, u)\,\mathrm{d}t \\ \ge& \frac{1}{2}\|u\|^{2}-\phi_{1}(T_{2}) \|u\|^{\gamma_{1}} - \phi _{2}(T_{2})\| u\|^{\gamma_{2}} \rightarrow+\infty \end{aligned}$$

as $\|u\|\rightarrow+\infty$ and $u\in E^{+}$, since $1<\gamma _{1}<\gamma_{2}<2$.

Step 3. Taking $e\in E^{+}$ with $\|e\|=1$, there exist $s_{0}\in(0, 1)$ and $\sigma_{0}>0$ such that

$$ \Phi(u)\le-\sigma_{0}, \quad \forall u\in S_{e}:=E^{-}\oplus E^{0}\oplus s_{0}e. $$

(3.23)

Set $X=E^{-}\oplus E^{0}\oplus \mathbb {R}e$. For $u=u^{-}+u^{0}+se\in X$, by (3.5), (3.7) and (W4),

$$\begin{aligned} \Phi(u) = & \frac{1}{2} \bigl(\|se\|^{2}- \bigl\| u^{-}\bigr\| ^{2} \bigr)-\int_{\mathbb {R}}W(t, u)\, \mathrm{d}t \\ \le& \frac{s^{2}}{2}-b_{3}\bigl\| u^{1} \bigr\| _{\gamma_{6}}^{\gamma_{6}}-b_{4}\bigl\| u^{2}\bigr\| _{\gamma_{7}}^{\gamma_{7}}. \end{aligned}$$

(3.24)

On the other hand, one sees that

$$\begin{aligned} s^{2}\|e\|_{2}^{2} = (se, se)_{2} = (se, u)_{2}=\bigl(se, u^{1}\bigr)_{2}+\bigl(se, u^{2}\bigr)_{2} \le |s| \bigl(\|e\|_{\gamma_{6}'}\bigl\| u^{1}\bigr\| _{\gamma_{6}}+\|e \|_{\gamma _{7}'}\bigl\| u^{2}\bigr\| _{\gamma_{7}} \bigr), \end{aligned}$$

where $\gamma_{6}'=\gamma_{6}/(\gamma_{6}-1) > \gamma_{6}$ and $\gamma _{7}'=\gamma _{7}/(\gamma_{7}-1) > \gamma_{7}$. Hence,

$$ s\le C_{6} \bigl(\bigl\| u^{1}\bigr\| _{\gamma_{6}}+ \min\bigl\{ \bigl\| u^{2}\bigr\| _{\gamma_{7}}, 1\bigr\} \bigr), \quad \forall s \in(0, 1). $$

(3.25)

Combining (3.24) with (3.25), we have

$$\begin{aligned} \Phi(u) \le& \frac{s^{2}}{2}-b_{3}\bigl\| u^{1} \bigr\| _{\gamma_{6}}^{\gamma _{6}}-b_{4}\bigl\| u^{2} \bigr\| _{\gamma_{7}}^{\gamma_{7}} \\ \le& \frac{s^{2}}{2}-\min\{b_{3}, b_{4}\} \bigl[ \bigl\| u^{1}\bigr\| _{\gamma _{6}}^{\gamma_{6}} + \bigl(\min\bigl\{ \bigl\| u^{2}\bigr\| _{\gamma_{7}}, 1\bigr\} \bigr)^{\gamma_{7}} \bigr] \\ \le& \frac{s^{2}}{2}-2^{1-\gamma_{6}}\min\{b_{3}, b_{4}\} \bigl[\bigl\| u^{1}\bigr\| _{\gamma_{6}} + \bigl(\min\bigl\{ \bigl\| u^{2}\bigr\| _{\gamma_{7}}, 1\bigr\} \bigr) \bigr]^{\gamma_{6}} \\ \le& \frac{s^{2}}{2}-2^{1-\gamma_{6}}\min\{b_{3}, b_{4}\}C_{6}^{-\gamma _{6}}s^{\gamma_{6}} \\ = & \frac{s^{2}}{2}-C_{7}s^{\gamma_{6}}, \quad \forall u=u^{-}+u^{0}+se\in X, s\in(0, 1), \end{aligned}$$

which implies that there exist $s_{0}\in(0, 1)$ and $\sigma_{0}>0$ such that (3.23) holds.

Step 4. If $E^{-}\oplus E^{0}=\{0\}$, then Lemmas 2.9 and 3.2, Steps 1-3 imply that Φ has a minimum (<0) which yields a homoclinic solution for system (1.1).

If $E^{-}\oplus E^{0}\ne\{0\}$, by Step 2, one can take $C_{8}>0$ and $r>s_{0}$ large such that

$$\Phi(u)\ge-C_{8}, \quad \forall u\in E^{+} $$

and

$$\Phi(u)\ge0, \quad \forall u\in E^{+} \mbox{ with } \|u\| \ge r. $$

Let $Q=B_{r}\cap E^{+}$. Since $S_{e}$ and ∂Q link, by Lemma 2.7, −Φ has a critical point $u^{*}\in E$ with $\Phi(u^{*})\le-\sigma_{0}$, which is a nontrivial homoclinic solution of system (1.1). □

Proof of Theorem 1.2

Set $X=E$, $X_{1}=E^{-}\oplus E^{0}$ and $X_{2}=E^{+}$. In view of Lemma 3.2 and Steps 1 and 2 in the proof of Theorem 1.1, $X=X_{1}\oplus X_{2}$, $\dim X_{1}<+\infty$, $\Phi\in C^{1}(X, \mathbb {R})$, Φ satisfies the (PS)-condition and is bounded from below on $X_{2}$. Obviously, (W1) and (W5) imply $\Phi(0)=0$ and Φ is even. Next, we prove that assumption (ii) in Lemma 2.8 holds.

Let $\tilde{X}\subset X$ be any finite dimensional subspace. Then there exist constants $c_{0}=c(\tilde{X})>0$ and $c_{*}=c(\tilde{X})>0$ such that

$$ c_{0}\|u\|\le\|u\|_{\gamma_{6}}, \|u \|_{\gamma_{7}}, \|u\|_{\infty}\le c_{*}\|u\|, \quad \forall u\in\tilde{X}. $$

(3.26)

Since $\gamma_{6}\ge\gamma_{7}$, it follows from (3.7) and (3.26) that

$$ \begin{aligned}[b] \|u\|_{\gamma_{6}}^{\gamma_{6}} & = \bigl\| u^{1} \bigr\| _{\gamma_{6}}^{\gamma_{6}}+\bigl\| u^{2}\bigr\| _{\gamma_{6}}^{\gamma_{6}} \le\bigl\| u^{1}\bigr\| _{\gamma_{6}}^{\gamma_{6}}+\|u\|_{\infty}^{\gamma_{6}-\gamma _{7}} \bigl\| u^{2}\bigr\| _{\gamma_{7}}^{\gamma_{7}} \\ & \le \bigl\| u^{1}\bigr\| _{\gamma_{6}}^{\gamma_{6}}+\bigl\| u^{2} \bigr\| _{\gamma_{7}}^{\gamma _{7}}, \quad \forall u\in\tilde{X}, c_{*}\|u\|< 1. \end{aligned} $$

(3.27)

From (3.5), (3.7), (3.26), (3.27) and (W4), one has

$$\begin{aligned} \Phi(u) = & \frac{1}{2} \bigl(\bigl\| u^{+}\bigr\| ^{2}- \bigl\| u^{-}\bigr\| ^{2} \bigr)-\int_{\mathbb {R}}W(t, u)\, \mathrm{d}t \\ \le& \frac{1}{2}\|u\|^{2}-b_{3}\bigl\| u^{1} \bigr\| _{\gamma_{6}}^{\gamma_{6}}-b_{4}\bigl\| u^{2} \bigr\| _{\gamma_{7}}^{\gamma_{7}} \\ \le& \frac{1}{2}\|u\|^{2}-\min\{b_{3}, b_{4}\}\|u\|_{\gamma_{6}}^{\gamma _{6}} \\ \le& \frac{1}{2}\|u\|^{2}-c_{0}^{\gamma_{6}} \min\{b_{3}, b_{4}\}\|u\| ^{\gamma_{6}},\quad \forall u \in\tilde{X}, c_{*}\|u\|< 1. \end{aligned}$$

Since $1< \gamma_{6} <2$, the above implies that there exist $\rho=\rho (b_{3}, b_{4}, c_{0})=\rho(\tilde{X})\in(0, c_{*}^{-1})$ and $\sigma=\sigma(b_{3}, b_{4}, c_{0})=\sigma(\tilde{X})>0$ such that

$$\Phi(u)\le0, \quad \forall u\in B_{\rho}\cap\tilde{X} ; \qquad \Phi(u) \le-\sigma, \quad \forall u\in\partial B_{\rho}\cap \tilde{X}. $$

Hence assumption (ii) in Lemma 2.8 holds. By Lemma 2.8, Φ has infinitely many (pairs) critical points which are homoclinic solutions for system (1.1). □

Proof of Theorem 1.3

In the proof of Theorem 1.1, assumption (W3) is used only in Step 1 to prove that a (PS)-sequence $\{u_{n}\}_{n\in{\mathbb{N}}}\subset E$ is bounded. Therefore, we only prove that any (PS)-sequence $\{u_{n}\}_{n\in{\mathbb{N}}}\subset E$ is also bounded by using (W3′) instead of (W3). From (3.5), (3.6) and (W3′), we have

$$\begin{aligned} \bigl\langle \Phi'(u_{n}), u_{n} \bigr\rangle -2\Phi(u_{n}) = & \int_{\mathbb {R}}\bigl[2W(t, u_{n})-\bigl(\nabla W(t, u_{n}), u_{n}\bigr) \bigr]\,\mathrm{d}t \\ \ge& b_{5}\int_{\mathbb {R}}\bigl|u_{n}(t)\bigr|^{\gamma_{8}} \,\mathrm{d}t - b_{6}\int_{\mathbb {R}}|u_{n}|^{\gamma_{9}} \,\mathrm{d}t \\ = & b_{5}\|u_{n}\|_{\gamma_{8}}^{\gamma_{8}}-b_{6} \|u_{n}\|_{\gamma_{9}}^{\gamma_{9}}. \end{aligned}$$

It follows that there exists a constant $C_{9}>0$ such that

$$ b_{5}\|u_{n}\|_{\gamma_{8}}^{\gamma_{8}}-b_{6} \|u_{n}\|_{\gamma_{9}}^{\gamma _{9}}\le C_{9}\bigl(1+ \|u_{n}\|\bigr). $$

(3.28)

Since $\dim(E^{-}\oplus E^{0})<+\infty$, there exists a constant $C_{10}>0$ such that

$$ \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}^{2} = \bigl(u_{n}^{-}+u_{n}^{0}, u_{n} \bigr)_{2} \le\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{\gamma_{8}'}\|u_{n}\|_{\gamma_{8}} \le C_{10} \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}\|u_{n}\|_{\gamma_{8}}, $$

(3.29)

where $\gamma_{8}'=\gamma_{8}/(\gamma_{8}-1)$. Combining (3.28) with (3.29), one has

$$ \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}\le C_{11}\bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| _{2}^{2}\le C_{12} \bigl(1+\|u_{n} \|^{2/\gamma_{8}} +\|u_{n}\|^{2\gamma_{9}/\gamma_{8}} \bigr). $$

(3.30)

From (3.5), (3.11) and (3.30), we obtain

$$\begin{aligned} \|u_{n}\|^{2} = & \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}+\bigl\| u_{n}^{+}\bigr\| ^{2} \\ = & \bigl\| u_{n}^{-}+u_{n}^{0} \bigr\| ^{2}+2\Phi(u_{n})+\bigl\| u_{n}^{-} \bigr\| ^{2}+2\int_{\mathbb {R}}W(t, u_{n})\, \mathrm{d}t \\ \le& 2C_{12} \bigl(1+\|u_{n}\|^{2/\gamma_{8}}+ \|u_{n}\|^{2\gamma _{9}/\gamma _{8}} \bigr)+2\Phi(u_{n}) \\ &{} +2\phi_{1}(T_{2})\|u_{n}\|^{\gamma_{1}}+2 \phi_{2}(T_{2})\|u_{n}\|^{\gamma _{2}} \\ \le& C_{13} \bigl(1+\|u_{n}\|^{\gamma_{1}}+ \|u_{n}\|^{\gamma_{2}}+\|u_{n}\| ^{2/\gamma_{8}}+ \|u_{n}\|^{2\gamma_{9}/\gamma_{8}} \bigr). \end{aligned}$$

Since $1<\gamma_{1}<\gamma_{2}<2$, $1< \gamma_{9} < \gamma_{8}<2$, it follows that $\{\|u_{n}\|\}$ is bounded. The proof is complete. □

Proof of Theorem 1.4

Set $X=E$, $X_{1}=E^{-}\oplus E^{0}$ and $X_{2}=E^{+}$. In view of Lemma 3.2 and Steps 1 and 2 in the proof of Theorem 1.1, $X=X_{1}\oplus X_{2}$, $\dim X_{1}<+\infty$, $\Phi\in C^{1}(X, \mathbb {R})$, Φ satisfies the (PS)-condition and is bounded from below on $X_{2}$. Obviously, (W1) and (W5) imply $\Phi(0)=0$ and Φ is even. Next, we prove that assumption (ii) in Lemma 2.8 holds.

Let $\tilde{X}\subset X$ be any finite dimensional subspace. Then there exist constants $c_{0}=c(\tilde{X})>0$ and $c_{*}=c(\tilde{X})>0$ such that

$$ c_{0}\|u\|\le\|u\|_{\gamma_{10}}, \|u \|_{\gamma_{11}}\le c_{*}\|u\|, \quad \forall u\in\tilde{X}. $$

(3.31)

From (3.5), (3.31) and (W4′), one has

$$\begin{aligned} \Phi(u) = & \frac{1}{2} \bigl(\bigl\| u^{+}\bigr\| ^{2}- \bigl\| u^{-}\bigr\| ^{2} \bigr)-\int_{\mathbb {R}}W(t, u)\, \mathrm{d}t \\ \le& \frac{1}{2}\|u\|^{2}-b_{7}\|u \|_{L^{\gamma_{10}}}^{\gamma_{10}} + b_{8}\|u\|_{L^{\gamma_{11}}}^{\gamma_{11}} \\ \le& \frac{1}{2}\|u\|^{2}-b_{7}c_{0}^{\gamma_{10}} \|u\|^{\gamma_{10}} + b_{8}c_{*}^{\gamma_{11}}\|u\|^{\gamma_{11}}, \quad \forall u\in\tilde{X}. \end{aligned}$$

Since $1< \gamma_{10}< \gamma_{11} <2$, the above implies that there exist $\rho=\rho(b_{7}, b_{8}, c_{0})=\rho(\tilde{X})>0$ and $\sigma=\sigma(b_{7}, b_{8}, c_{0})=\sigma(\tilde{X})>0$ such that

$$\Phi(u)\le0, \quad \forall u\in B_{\rho}\cap\tilde{X} ; \qquad \Phi(u) \le-\sigma, \quad \forall u\in\partial B_{\rho}\cap \tilde{X}. $$

Hence assumption (ii) in Lemma 2.8 holds. By Lemma 2.8, Φ has infinitely many (pairs) critical points which are homoclinic solutions for system (1.1). □

In the proof of Theorem 1.4, (W4′) is used in the last part to verify assumption (ii) of Lemma 2.8. It is easy to see that it also holds by using (W4″) instead of (W4′). So we omit the proof of Corollary 1.5.

4 Examples

In this section, we give two examples to illustrate our results.

Example 4.1

In system (1.1), let $L(t)= (|t|^{4/5}-1 )I_{N}$, and $W(t, x)$ be as in (1.4). Then $L(t)$ satisfies ($\mathrm{L}_{\nu}$) with $\nu =6/5$, and

$$\begin{aligned}& \nabla W(t, x)= \bigl(1+\sin^{2}t \bigr) \biggl(\frac {5}{4}|x|^{-3/4}x- \frac {9}{2}|x|^{-1/2}x+\frac{7}{4}|x|^{-1/4}x \biggr), \\& \bigl|W(t, x)\bigr|\le5 \bigl(|x|^{5/4}+ |x|^{7/4} \bigr), \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}, \\& \bigl|\nabla W(t, x)\bigr|\le\frac{5|x|^{1/4}+18|x|^{1/2}+7|x|^{3/4}}{2}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}, \\& 2W(t, x)-\bigl(\nabla W(t, x), x\bigr) \ge\frac{1}{4}|x|^{7/4} -3 |x|^{3/2}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N} \end{aligned}$$

and

$$W(t, x)\ge|x|^{5/4} -6 |x|^{3/2}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}. $$

Thus all conditions of Theorem 1.4 are satisfied with

$$\begin{aligned}& \frac{5}{4}=\gamma_{1}=\gamma_{3}= \gamma_{10}< \gamma_{9}=\gamma _{11}= \frac {3}{2}< \gamma_{8}=\gamma_{2}=\frac{7}{4}; \qquad a_{1}=a_{2}=5; \\& b_{5}=\frac{1}{4},\qquad b_{6}=3, \qquad b_{7}=1,\qquad b_{8}=6; \qquad \varphi (s)= \frac{5s^{1/4}+18s^{1/2}+7s^{3/4}}{2}. \end{aligned}$$

Hence, by Theorem 1.4, system (1.1) has infinitely many nontrivial homoclinic solutions.

Example 4.2

In system (1.1), let $L(t)= (|t|^{\varrho }-1 )I_{N}$, let $W(t, x)$ be as in (1.5). Set

$$\begin{aligned}& \lambda_{i}=\frac{\tau_{m}-\tau_{i}}{\tau_{m}-\tau_{1}}, \qquad \mu _{i}= \frac{\tau _{i}-\tau_{1}}{\tau_{m-1}-\tau_{1}},\qquad \theta_{j}=\frac{\tau_{m}-\tau_{j}}{\tau_{m}-\tau_{2}}, \quad i=2, \ldots, m-1; j=3, \ldots, m-1; \\& a_{1}=d_{1}+\sum_{i=2}^{m-1} \lambda_{i}d_{i}, \qquad a_{2}=d_{m}+ \sum_{i=2}^{m-1}(1-\lambda_{i})d_{i}; \\& b_{6}=(2-\tau_{m-1})d_{m-1}+\sum _{i=3}^{m-2}(2-\tau_{i}) \mu_{i}d_{i} \bigl[m(2-\tau_{i})d_{i} (1-\mu_{i} ) \bigr]^{(1-\mu_{i})/\mu_{i}} |x|^{\tau_{m-1}} \end{aligned}$$

and

$$b_{8}=d_{2}+\sum_{j=3}^{m-1} \theta_{j}d_{j} \bigl[md_{j} (1-\theta _{j} ) \bigr]^{(1-\theta_{j})/\theta_{j}}. $$

Note that

$$\begin{aligned}& |x|^{\tau_{i}}\le\lambda_{i}|x|^{\tau_{1}}+ (1- \lambda_{i} )|x|^{\tau _{m}}, \quad i=2, \ldots, m-1, \\& |x|^{\tau_{i}}\le\frac{1}{m(2-\tau_{i})d_{i}}|x|^{\tau_{1}}+\mu_{i} \bigl[m(2-\tau _{i})d_{i} (1-\mu_{i} ) \bigr]^{(1-\mu_{i})/\mu_{i}} |x|^{\tau_{m-1}}, \quad i=2, \ldots, m-2 \end{aligned}$$

and

$$|x|^{\tau_{j}}\le\theta_{j} \bigl[md_{j} (1- \theta_{j} ) \bigr]^{(1-\theta_{j})/\theta_{j}}|x|^{\tau_{2}} + \frac{1}{md_{j}}|x|^{\tau_{m}}, \quad j=3, \ldots, m-1. $$

Then $L(t)$ satisfies ($\mathrm{L}_{\nu}$) with $\nu=2-\varrho<3-2/\tau_{1}$, and

$$\begin{aligned}& \bigl|W(t, x)\bigr| \le a_{1}|x|^{\tau_{1}}+ a_{2}|x|^{\tau_{m}}, \qquad \bigl|\nabla W(t, x)\bigr| \le\sum_{i=1}^{m} \tau_{i}d_{i}|x|^{\tau_{i}-1},\quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}, \\& 2W(t, x)-\bigl(\nabla W(t, x), x\bigr) \ge(2-\tau_{m})d_{m}|x|^{\tau_{m}} -b_{6}|x|^{\tau _{m-1}}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N} \end{aligned}$$

and

$$W(t, x)\ge d_{1}|x|^{\tau_{1}} -b_{8} |x|^{\tau_{2}}, \quad \forall (t, x)\in \mathbb {R}\times{ \mathbb {R}}^{N}. $$

Theorem 1.4 applies with

$$\begin{aligned}& \tau_{1}=\gamma_{1}=\gamma_{3}= \gamma_{10}< \gamma_{11}=\tau_{2}< \gamma _{9}=\tau _{m-1}< \gamma_{8}= \gamma_{2}=\tau_{m}; \\& b_{5}=(2-\tau_{m})d_{m}, \qquad b_{7}=d_{1};\qquad \varphi(s)=\sum_{i=1}^{m} \tau _{i}d_{i}s^{\tau_{i}-1}, \end{aligned}$$

and system (1.1) has infinitely many nontrivial homoclinic solutions.