Ground State Homoclinic Orbits for First-Order Hamiltonian System

Abstract

In this paper, we study the following first-order Hamiltonian system

$$\begin{aligned} \dot{z}=\mathscr {J}H_{z}(t,z). \end{aligned}$$

Under some suitable conditions on the nonlinearity, we establish the existence of ground state homoclinic orbits by using variational methods. Moreover, we also explore some properties of these homoclinic orbits, such as compactness of set of and exponential decay of ground state homoclinic orbits.

1 Introduction and Main Results

We study the following first-order Hamiltonian system

$$\begin{aligned} \dot{z}=\mathscr {J}H_{z}(t,z) \end{aligned}$$

(1.1)

where $z=(p,q)\in \mathbb {R}^{N}\times \mathbb {R}^{N}=\mathbb {R}^{2N}$, and

$$\begin{aligned} \mathscr {J}=\left( \begin{array}{cc} 0 &{} -\,I_{N} \\ I_{N} &{} 0 \\ \end{array} \right) \end{aligned}$$

is the standard symplectic matrix with $I_{N}$ being the identity matrix in $\mathbb {R}^{N}$. Here the Hamiltonian function has the form

$$\begin{aligned} H(t,z)=\frac{1}{2}L(t)z\cdot z+G(t,z) \end{aligned}$$

with L being a symmetric $2N\times 2N$ matrix-valued function. In this paper, we are concerned with the existence of ground state homoclinic orbits and show some properties of homoclinic orbits. As usual, we say that a solution z of system (1.1) is a homoclinic orbit if $z(t)\not \equiv 0$ and $z(t)\rightarrow 0$ as $|t|\rightarrow \infty$.

As a special case of dynamical systems, Hamiltonian systems are very important in the study of gas dynamics, finance, fluid mechanics, relativistic mechanics and nuclear physics (see [1, 17]). By using modern variational methods, many authors were devoted to the existence of periodic and homoclinic solutions for Hamiltonian systems in the literature over the past several decades. More precisely, according to the various hypotheses on the functions L and G, the existence and multiplicity of homoclinic orbits of system (1.1) have been established by many authors. For example, see [2,3,4, 6,7,8,9,10,11, 13, 18, 20, 22, 23, 26, 30, 31, 33, 34] and their references. For readers’ convenience, next we briefly recall some of them as follows.

Most of the results focused on the case G(t, z) are autonomous or depend periodically on t. As far as variational methods are concerned, there is a pioneering work by Coti-Zelati, Ekeland and Séré [2] where they first proved the existence and multiplicity of homoclinic orbits by mountain pass argument under strictly convexity and Ambrosetti–Rabinowitz growth condition. This result was deepened in [20] where the existence of infinitely many homoclinic orbits was obtained. Independently, without the convexity condition, the existence result of homoclinic orbit was obtained in [13] by Fredholm operator theory and a linking argument and in [26] by a subharmonic approach. Later variational methods with linking type arguments were used in [4, 6, 7, 11, 22, 23] to show the existence and multiplicity of homoclinic orbits of system (1.1) under some general growth conditions for the nonlinearity G(t, z). For the super-quadratic periodic case, see [4, 6, 7, 11], and for the asymptotically quadratic periodic case, we refer readers to [6, 22, 23]. It is worth pointing out that the periodicity is used to control the lack of compactness due to the fact that system (1.1) is set on all $\mathbb {R}$.

On the other hand, system (1.1) with nonperiodic condition has been attracted attention in recent years. However, without the assumption of periodicity, the problem is quite different in nature due to the lack of compactness of Sobolev embedding. In the early paper [9], Ding and Li first studied the nonperiodic problem and obtained the existence of homoclinic orbits for (1.1) under some fairly strong assumptions on L. Subsequently, Ding and Jeanjean [8] obtained the existence and multiplicity of homoclinic orbits for (1.1) by assuming that L satisfies a more general condition than those in [9] and G(t, z) is asymptotically quadratic at infinity. After the work of [9], there are also a few papers devoted to the nonperiodic problem; see [10, 30, 33, 34] for the super-quadratic growth case and [31] for the sub-quadratic growth case. It should be noted that the above-mentioned works mainly take advantage of the properties of L to recover sufficient compactness.

Motivated by the above facts, in this paper we are interested in the nonlinearity G is asymptotically periodic at infinity. To the best of our knowledge, there are no works which deal with the asymptotically periodic case for system (1.1). More specifically, we will prove the existence of ground state homoclinic orbits of system (1.1) by applying variational methods and analyze some properties of ground state homoclinic orbits by using some analysis techniques, such as compactness of set of and exponential decay of ground state homoclinic orbits.

In order to precisely state our results, we denote by $\mathscr {K}$ the class of functions $k\in C(\mathbb {R})\cap L^{\infty }(\mathbb {R})$ satisfying, for any $\epsilon >0$, the set $\left\{ t\in \mathbb {R}: |k(t)|\ge \epsilon \right\}$ has finite Lebesgue measure. We also denote by $\sigma (A)$ the spectrum of operator A. Moreover, we assume that L and G satisfy the following conditions:

(L):

L is a constant symmetric $2N\times 2N$ matrix such that $\sigma (\mathscr {J}L)\cap i\mathbb {R}=\emptyset$;

$(g_{1})$:

$G_{z}(t,z)=g(t,|z|)z$, where $g\in C(\mathbb {R}\times \mathbb {R}^{+}, \mathbb {R}^{+})$, and there exist $p>2$ and $c_{0}>0$ such that

$$\begin{aligned} |g(t,s)|\le c_{0}(1+|s|^{p-2})~~\hbox {for all}~(t,s)\in \mathbb {R}\times \mathbb {R}^{+}; \end{aligned}$$

$(g_{2})$:

$g(t,s)=o(1)$ as $s\rightarrow 0$ uniformly in t;

$(g_{3})$:

$\frac{G(t,s)}{s^{2}}\rightarrow \infty$ as $s\rightarrow \infty$ uniformly in t;

$(g_{4})$:

g(t, s) is nondecreasing in s on $(0,+\,\infty )$;

$(g_{5})$:

there exist constant $q>2$ and function $k\in \mathscr {K}$, $g_{0}\in C(\mathbb {R}\times \mathbb {R}^{+}, \mathbb {R}^{+})$ with 1-periodic in t such that

• (i) $G(t, s)> G_{0}(t, s)=\int _{0}^{s}g_{0}(t, \tau )\tau d\tau$ for all $(t, s)\in \mathbb {R}\times \mathbb {R}^{+}$;

• (ii) $|g(t, s)-g_{0}(t, s)|\le k(t)(1+|s|^{q-2})$ for all $(t, s)\in \mathbb {R}\times \mathbb {R}^{+}$;

• (iii) $g_{0}(t, s)$ is nondecreasing in s on $(0,+\,\infty )$.

We are now in a position to state the main results of this paper.

Theorem 1.1

Suppose that (L) and $(g_{1})$–$(g_{5})$ are satisfied. Then

• (1) system (1.1) has at least a ground state homoclinic orbit;

• (2) $\mathscr {L}$ is compact in $H^{1}(\mathbb {R},\mathbb {R}^{2N})$, where $\mathscr {L}$ denotes the set of all ground state homoclinic orbits of system (1.1);

• (3) in addition, if $g\in C^{1}$ satisfies $g_{t}(t,s)=o(1)$ and $g_{s}(t,s)s=o(1)$ as $s\rightarrow 0$ uniformly in t, then there exist constants $c, C>0$ such that

  $$\begin{aligned} |z(t)|\le C\exp \left( -\,c|t|\right) ~\hbox {for all}~t\in \mathbb {R},z\in \mathscr {L}. \end{aligned}$$

Next we give the main ideas for the proof of the main results. Our argument is variational, which can be outlined as follows. The solutions are obtained as critical points of the energy functional associated with system (1.1). Roughly speaking, because the energy functional of system (1.1) is strongly indefinite, we shall use generalized linking theorem [14, 16] and the diagonal method [27, 28] to construct a Cerami sequence for the energy functional at some level. After proving that the Cerami sequence is bounded, we show that its weak limit is a ground state homoclinic orbits of system (1.1). However, since system (1.1) is set on all $\mathbb {R}$, the main difficulty when dealing with this problem is the lack of compactness of Sobolev embedding. It is natural to ask how to show this weak limit is nontrivial? Moreover, for the asymptotically periodic problem, the energy functional loses the translation invariance, which brings new difficulty. For these reasons, many effective methods for periodic problem cannot be applied to asymptotically periodic problem. Fortunately, with the help of the limit problem, we may take advantage of the asymptotic property of the nonlinearity to establish a global compactness result for bounded Cerami sequences in order to recover sufficient compactness. Finally, by some analysis techniques, we show some properties of ground state homoclinic orbits.

The remainder of this paper is organized as follows. In Sect. 2, we formulate the variational setting and introduce some useful preliminaries. We complete the proofs of main results in Sect. 3.

2 Variational Setting and Preliminaries

Below by $|\cdot |_{q}$, we denote the usual $L^{q}$- norm, $(\cdot ,\cdot )_{2}$ denote the usual $L^{2}$ inner product, and $c,c_{i}$ or $C_{i}$ stand for different positive constants. For convenience, we need the following notations. Let Hamiltonian operator

$$\begin{aligned} A=-\,\left( \mathscr {J}\frac{d}{dt}+L\right) . \end{aligned}$$

Observe that (L) means L is independent of t and there is $\alpha >0$ such that $(-\alpha ,\alpha )\cap \sigma (A)=\emptyset$ (see [6, 22]). Then the space $L^{2}:=L^{2}(\mathbb {R},\mathbb {R}^{2N})$ possesses the orthogonal decomposition

$$\begin{aligned} L^{2}=L^{-}\oplus L^{+},~~~z=z^{-}+z^{+} \end{aligned}$$

such that A is negative definite (resp. positive definite) in $L^{-}$(resp. $L^{+}$). Let |A| denote the absolute value of A and $|A|^{1/2}$ be the square root of |A|. Let $E=\mathcal {D}(|A|^{1/2})$ be the Hilbert space with the inner product

$$\begin{aligned} (z,w)=(|A|^{1/2}z, |A|^{1/2}w)_{2} \end{aligned}$$

and norm $\Vert z\Vert =(z,z)^{1/2}$. Moreover, it is clear that E possesses the following decomposition

$$\begin{aligned} E=E^{-}\oplus E^{+},~~\hbox {where}~~E^{\pm }=E\cap L^{\pm }, \end{aligned}$$

which is orthogonal with respect to the inner products $(\cdot ,\cdot )_{2}$ and $(\cdot ,\cdot )$. Moreover, E embeds continuously into $L^{q}$ for all $q\ge 2$ and compactly into $L^{q}_\mathrm{loc}$ for all $q\ge 1$; then, there exists constant $\gamma _{p}>0$ such that for all $z\in E$

$$\begin{aligned} |z|_{p}\le \gamma _{p}\Vert z\Vert ~\hbox {for all}~p\ge 2. \end{aligned}$$

(2.1)

In virtue of the assumptions $(g_{1})$ and $(g_{2})$, for any $\epsilon >0$, there exists positive constant $C_{\epsilon }$ such that

$$\begin{aligned} g(t,s)\le \epsilon +C_{\epsilon }|s|^{p-2}~~\hbox {and}~~ |G(t,s)|\le \epsilon |s|^{2}+C_{\epsilon }|s|^{p} \end{aligned}$$

(2.2)

for all $(t,s)\in \mathbb {R}\times \mathbb {R}^{+}$ and $p>2$. By $(g_{4})$, we obtain

$$\begin{aligned} \frac{1}{2}g(t,s)s^{2}\ge G(t,s)\ge 0,~\hbox {for all}~(t,s)\in \mathbb {R}\times \mathbb {R}^{+}. \end{aligned}$$

(2.3)

Now we define the energy functional of system (1.1) on E by

$$\begin{aligned} \Phi (z)= \frac{1}{2}(\Vert z^{+}\Vert ^{2}-\Vert z^{-}\Vert ^{2}) -\int _{\mathbb {R}}G(t,|z|). \end{aligned}$$

Obviously, the energy functional $\Phi$ is strongly indefinite. Moreover, our hypotheses imply that $\Phi \in C^{1}(E, \mathbb {R})$, and

$$\begin{aligned} \Phi ^\prime (z)\varphi =(z^+,\varphi ^+)-(z^-,\varphi ^-)-\int _{\mathbb {R}}g(t,|z|)z\cdot \varphi ,~\forall \varphi \in E, \end{aligned}$$

where the dot stands for the scalar product in $\mathbb {R}^{2N}$, and a standard argument shows that critical points of $\Phi$ are solutions of system (1.1) (see [5, 29]).

In order to seek for the ground state homoclinic orbits of system (1.1), we consider the following set which is introduced in [19, 24]

$$\begin{aligned} \mathcal {M}:=\{z\in E\backslash E^{-}: \Phi '(z)z=0~\hbox {and}~\Phi '(z)w=0~\hbox {for any}~w\in E^{-}\}. \end{aligned}$$

Following from Szulkin and Weth [24], we will call the set $\mathcal {M}$ the generalized Nehari manifold. Clearly, the set $\mathcal {M}$ is a natural constraint and it contains all nontrivial critical points of $\Phi$. Let

$$\begin{aligned} m:=\inf _{z\in \mathcal {M}}\Phi . \end{aligned}$$

If m is attained by $z_{0}\in \mathcal {M}$, then $z_{0}$ is a critical point of $\Phi$. Since m is the lowest level for $\Phi$, $z_{0}$ be called a ground state homoclinic orbit of system (1.1).

Recall that for a functional $\Phi \in C^{1}(E,\mathbb {R})$, $\Phi$ is said to be weakly sequentially lower semi-continuous if for any $u_{n}\rightharpoonup u$ in E one has $\Phi (u)\le \liminf \limits _{n\rightarrow \infty }\Phi (u_{n})$, and $\Phi ^\prime$ is said to be weakly sequentially continuous if $\lim \limits _{n\rightarrow \infty }\Phi ^\prime (u_n)v=\Phi ^\prime (u)v$ for each $v\in E$. We recall that a sequence $\{u_{n}\}\subset E$ is called Cerami sequence for $\Phi$ at the level c ($(C)_{c}$-sequence in short) if

$$\begin{aligned} \Phi (u_{n})\rightarrow c~~\hbox {and}~~(1+\Vert u_{n}\Vert )\Vert \Phi ^\prime (u_{n})\Vert \rightarrow 0. \end{aligned}$$

We say that $\Phi$ satisfy the $(C)_{c}$-condition if any $(C)_{c}$-sequence has a convergent subsequence in E.

To prove the main result, we need the following generalized linking theorem due to [14, 16].

Lemma 2.1

Let X be a real Hilbert space with $X=X^{-}\oplus X^{+}$, and let $\Phi \in C^{1}(X, {\mathbb {R}})$ be of the form

$$\begin{aligned} \Phi (u)=\frac{1}{2}\left( \Vert u^{+}\Vert ^2-\Vert u^{-}\Vert ^2\right) -\Psi (u), \ \ \ \ u=u^{-}+u^{+}\in X^{-}\oplus X^{+}. \end{aligned}$$

Suppose that the following assumptions are satisfied:

$(\Psi _1)$:

$\Psi \in C^{1}(X, {\mathbb {R}})$ is bounded from below and weakly sequentially lower semi-continuous;

$(\Psi _2)$:

$\Psi '$ is weakly sequentially continuous;

$(\Psi _3)$:

there exist $R>\rho >0$ and $e\in X^{+}$ with $\Vert e\Vert =1$ such that

$$\begin{aligned} \kappa :=\inf \Phi (S^{+}_{\rho }) > \sup \Phi (\partial Q), \end{aligned}$$

where

$$\begin{aligned} S^{+}_{\rho }=\left\{ u\in X^{+} : \Vert u\Vert =\rho \right\} , \ \ \ \ Q=\left\{ v+se : v\in X^{-},\ s\ge 0,\ \Vert v+se\Vert \le R\right\} . \end{aligned}$$

Then there exist a constant $c\in [\kappa , \sup \Phi (Q)]$ and a sequence $\{u_n\}\subset X$ satisfying

$$\begin{aligned} \Phi (u_n)\rightarrow c~~\hbox {and}~~(1+\Vert u_n\Vert )\Vert \Phi ^\prime (u_n)\Vert \rightarrow 0. \end{aligned}$$

For the sake of convenience, we write

$$\begin{aligned} \Psi (z)=\int _{\mathbb {R}}G(t,|z|). \end{aligned}$$

Employing a standard argument, one can check easily the following lemma (see [5]).

Lemma 2.2

Suppose that $(g_{1})$–$(g_{4})$ are satisfied. Then $\Psi$ is nonnegative, weakly sequentially lower semi-continuous, and $\Psi '$ is weakly sequentially continuous.

Applying some arguments in [35], we obtain the following important estimate, and the proof can be found in [35] (see also [36]).

Lemma 2.3

Suppose that (L) and $(g_{1})$–$(g_{4})$ are satisfied. Let $z\in E$, $w\in E^{-}$ and $r\ge 0$. Then

$$\begin{aligned} \Phi (z)\ge \Phi (rz+w)-\Phi '(z)\left( \frac{r^{2}-1}{2}z+rw\right) . \end{aligned}$$

(2.4)

As a consequence of Lemma 2.3, we have the following lemma.

Lemma 2.4

Suppose that (L) and $(g_{1})$–$(g_{4})$ are satisfied. Then for $z\in \mathcal {M}$, $w\in E^{-}$ and $r\ge 0$

$$\begin{aligned} \Phi (z)\ge \Phi (rz+w). \end{aligned}$$

For the need later, we write $E(z):=E^{-}\oplus \mathbb {R}^{+}z$ for $z\in E\backslash E^{-}$. Let $z\in \mathcal {M}$, then Lemma 2.4 implies that z is the global maximum of $\Phi |_{E(z)}$.

Lemma 2.5

Suppose that (L) and $(g_{1})$–$(g_{4})$ are satisfied. Then

• (i) there exists $\rho >0$ such that

  $$\begin{aligned} m=\inf _{\mathcal {M}}\Phi \ge \kappa :=\inf _{S_{\rho }}\Phi >0, \end{aligned}$$

  where $S_{\rho }:=\{z\in E^{+}, \Vert z\Vert =\rho \}$.

• (ii) $\Vert z^{+}\Vert \ge \max \left\{ \Vert z^{-}\Vert , \sqrt{2m}\right\}$ for all $z\in \mathcal {M}$.

Proof

• (i) For $z\in E^{+}$, by (2.1) and (2.2), we obtain

  $$\begin{aligned} \Phi (z)=\frac{1}{2}\Vert z\Vert ^{2}-\int _{\mathbb {R}}G(t,|z|)\ge \left( \frac{1}{2}-\epsilon \gamma _{2}^{2}\right) \Vert z\Vert ^{2} -\,\gamma _{p}^{p}C_{\epsilon }\Vert z\Vert ^{p}. \end{aligned}$$

  It is easy to see that there exists $\rho >0$ small enough such that $\kappa :=\inf _{S_{\rho }}\Phi >0$. So the second inequality holds. Note that for every $z\in \mathcal {M}$ there is $s>0$ such that $sz^{+}\in E(z)\cap S_{\rho }$. Clearly, the first inequality follows from Lemma 2.4.

• (ii) or $z\in \mathcal {M}$, by (2.3) we have

  $$\begin{aligned} m\le \frac{1}{2}(\Vert z^{+}\Vert ^{2}-\Vert z^{-}\Vert ^{2})-\int _{\mathbb {R}}G(t,|z|)\le \frac{1}{2}(\Vert z^{+}\Vert ^{2}-\Vert z^{-}\Vert ^{2}); \end{aligned}$$

  hence, $\Vert z^{+}\Vert \ge \max \left\{ \Vert z^{-}\Vert , \sqrt{2m}\right\}$.

$\square$

Lemma 2.6

Suppose that (L) and $(g_{1})$–$(g_{4})$ are satisfied. Then for any $e\in E^{+}$ with $\Vert e\Vert =1$, $\sup \Phi (E^{-}\oplus {\mathbb {R}}^{+} e)<\infty$, and there is $R_e>0$ such that

$$\begin{aligned} \Phi (z) < 0,~~ \forall \ z\in E^{-}\oplus \mathbb {R}^{+} e, ~~\Vert z\Vert \ge R_e. \end{aligned}$$

In particular, there is a $R_0>\rho$ such that $\sup \Phi (\partial Q_{R})\le 0$ for $R\ge R_0$, where

$$\begin{aligned} Q_{R}=\left\{ se+w: w\in E^{-}, \ s\ge 0, \ \Vert se+w\Vert \le R\right\} . \end{aligned}$$

(2.5)

Proof

It is sufficient to prove that $\Phi (z)\rightarrow -\,\infty$ as $\Vert z\Vert \rightarrow +\,\infty$ for $z\in E^{-}\oplus {\mathbb {R}}^{+} e$. If not, then there are constant $M>0$ and a sequence $\{z_n\}\subset E^{-}\oplus {\mathbb {R}}^{+} e$ with $\Vert z_{n}\Vert \rightarrow +\,\infty$ such that $\Phi (z_{n})>-M$ for all n. Denote $w_n=z_n/\Vert z_n\Vert =s_ne+w_{n}^{-}$, where $s_n\ge 0$, $w^{-}_n\in E^{-}$, then $1=\Vert w_n\Vert ^{2}=s_n^{2}+\Vert w^{-}_n\Vert ^{2}$. Passing to a subsequence if necessary, $w_n\rightharpoonup w$ in E, $w_{n}^{-}\rightharpoonup w^{-}$ in $E^{-}$, $w_{n}(t)\rightarrow w(t)$ a.e. on $\mathbb {R}$, and $s_{n}\rightarrow s\ge 0$. Observe that by (2.3) we get

$$\begin{aligned} \frac{-M}{\Vert z_n\Vert ^{2}}\le \frac{\Phi (z_{n})}{\Vert z_n\Vert ^{2}}=\frac{1}{2}(s_{n}^{2}-\Vert w^{-}_{n}\Vert ^{2}) -\int _{\mathbb {R}}\frac{G(t,|z_{n}|)}{\Vert z_n\Vert ^{2}} \le \frac{1}{2}\left( s_{n}^{2}-\Vert w_{n}^-\Vert ^{2}\right) . \end{aligned}$$

If $s=0$, then $\Vert w^{-}_{n}\Vert \rightarrow 0$; this contradicts with $\Vert w_{n}\Vert =1$. So $s>0$ and $w=se+w^-\ne 0$. Then $|z_{n}|=\Vert z_n\Vert |w_n|\rightarrow +\,\infty$. By $(g_{3})$, (2.3) and Fatou’s lemma, we get

$$\begin{aligned} \begin{aligned} 0&\le \lim _{n\rightarrow +\,\infty }\frac{\Phi (z_{n})}{\Vert z_n\Vert ^{2}} =\lim _{n\rightarrow +\,\infty }\left( \frac{1}{2}(s_{n}^{2}-\Vert w^{-}_{n}\Vert ^{2}) -\int _{\mathbb {R}}\frac{G(t,|z_{n}|)}{\Vert z_n\Vert ^{2}}\right) \\&\le \frac{1}{2}s^2 -\liminf _{n\rightarrow +\infty }\int _{\mathbb {R}}\frac{G(t,|z_{n}|)}{|z_n|^{2}}|w_n|^{2}\\&\le \frac{1}{2}s^2-\int _{\mathbb {R}}\liminf _{n\rightarrow +\infty }\frac{G(t,|z_{n}|)}{|z_n|^{2}}|w_n|^{2}\\&=-\,\infty . \end{aligned} \end{aligned}$$

This is a contradiction. The proof is completed. $\square$

As a consequence of Lemmas 2.1, 2.2, 2.5 and 2.6, we have

Lemma 2.7

Suppose that (L) and $(g_{1})$–$(g_{4})$ are satisfied. Then there exist a constant $\hat{c}\in [\kappa , \sup \Phi (Q)]$ and a sequence $\{z_n\}\subset E$ satisfying

$$\begin{aligned} \Phi (z_n)\rightarrow \hat{c}~~\hbox {and}~~ \Vert \Phi '(z_n)\Vert (1+\Vert z_n\Vert )\rightarrow 0. \end{aligned}$$

Next we construct a $(C)_{\tilde{c}}$-sequence for some $\tilde{c}\in [\kappa , m]$ via a diagonal method developed by Tang in [27, 28], which is very important in our arguments.

Lemma 2.8

Suppose that (L) and $(g_{1})$–$(g_{4})$ are satisfied. Then there exist a constant $\tilde{c}\in [\kappa , m]$ and a sequence $\{z_n\}\subset E$ satisfying

$$\begin{aligned} \Phi (z_n)\rightarrow \tilde{c}~~\hbox {and}~~\Vert \Phi ^\prime (z_n)\Vert (1+\Vert z_n\Vert )\rightarrow 0. \end{aligned}$$

Proof

Choose $\xi _k\in \mathcal {M}$ such that

$$\begin{aligned} m\le \Phi (\xi _k)< m+\frac{1}{k},~~ k\in \mathbb {N}. \end{aligned}$$

(2.6)

By Lemma 2.5-(ii), $\Vert \xi _k^{+}\Vert \ge \sqrt{2m}>0$. Set $e_k=\xi _k^{+}/\Vert \xi _k^{+}\Vert$. Then $e_k\in E^{+}$ and $\Vert e_k\Vert =1$. In view of Lemma 2.6, there exists $R_k>\max \{\rho ,\Vert \xi _k\Vert \}$ such that $\sup \Phi (\partial Q_k)\le 0$, where

$$\begin{aligned} Q_k=\{se_k+w: w\in E^{-}, \ s\ge 0, \ \Vert se_k+w\Vert \le R_k\},~~ k\in \mathbb {N}. \end{aligned}$$

(2.7)

Hence, using Lemma 2.7 to the above set $Q_k$, there exist a constant $c_{k}\in [\kappa , \sup \Phi (Q_k)]$ and a sequence $\{z_{k, n}\}_{n\in \mathbb {N}}\subset E$ satisfying

$$\begin{aligned} \Phi (z_{k, n})\rightarrow c_{k}~~\hbox {and}~~\Vert \Phi '(z_{k, n})\Vert (1+\Vert z_{k, n}\Vert )\rightarrow 0,~~ k\in \mathbb {N}. \end{aligned}$$

(2.8)

By virtue of Lemma 2.4, one can get that

$$\begin{aligned} \Phi (\xi _k) \ge \Phi (t\xi _k+w), ~~ \forall \ t\ge 0,~ w\in E^{-}. \end{aligned}$$

(2.9)

Since $\xi _k\in Q_k$, it follows from (2.7) and (2.9) that $\Phi (\xi _k)=\sup \Phi (Q_k)$. Hence, by (2.6) and (2.8), one has

$$\begin{aligned} \Phi (z_{k, n})\rightarrow c_{k}< m+\frac{1}{k}~~\hbox {and}~~\Vert \Phi '(z_{k, n})\Vert (1+\Vert z_{k, n}\Vert )\rightarrow 0,~~ k\in \mathbb {N}. \end{aligned}$$

Now, we can choose a sequence $\{n_k\}\subset \mathbb {N}$ such that

$$\begin{aligned} \Phi (z_{k, n_k})<m+\frac{1}{k}~~\hbox {and}~~\Vert \Phi '(z_{k, n_k})\Vert (1+\Vert z_{k, n_k}\Vert )<\frac{1}{k},~~ k\in \mathbb {N}. \end{aligned}$$

Let $z_k=z_{k, n_k}, k\in \mathbb {N}$. Then, going if necessary to a subsequence, we have

$$\begin{aligned} \Phi (z_k)\rightarrow \tilde{c}\in [\kappa , m]~~\hbox {and}~~ \Vert \Phi '(z_k)\Vert (1+\Vert z_k\Vert )\rightarrow 0. \end{aligned}$$

The proof is completed. $\square$

In a similar way to [24, Lemma 2.6], we have

Lemma 2.9

Suppose that (L) and $(g_{1})$–$(g_{4})$ are satisfied. Then for any $z\in E\setminus E^{-}$, $\mathcal {M}\cap E(z)\ne \emptyset$, i.e., there exist $t>0$ and $w\in E^{-}$ such that $tz+w\in \mathcal {M}$.

Proof

Since $E(z)=E^{-}\oplus \mathbb {R}^{+}z=E^{-}\oplus \mathbb {R}^{+}z^{+}=E(z^{+})$, we may assume that $z\in E^+$. By Lemma 2.6, there exists $R>0$ such that $\Phi (z)\le 0$ for $z\in E(z)\backslash B_{R}(0)$. By Lemma 2.5-(i), $\Phi (tz)>0$ for small $t>0$. Thus, $0<\sup \Phi (E(z))<\infty$. It is easy see that $\Phi$ is weakly upper semi-continuous on E(z); therefore, $\Phi (z_0)=\sup \Phi (E(z))$ for some $z_0\in E(z)$. This $z_0$ is a critical point of $\Phi |_{E(z)}$, so $\Phi '(z_{0})z_{0}=\Phi '(z_{0})w=0$ for all $w\in E(z)$. Consequently, $z_{0}\in \mathcal {M}\cap E(z)$. $\square$

Next we discuss the behavior of the $(C)_{c}$-sequence.

Lemma 2.10

Suppose that (L) and $(g_{1})$–$(g_{4})$ are satisfied. Then any $(C)_{c}$-sequence of $\Phi$ at level $c\ge 0$ is bounded.

Proof

Let $\{z_{n}\}\subset E$ be such that

$$\begin{aligned} (1+\Vert z_{n}\Vert )\Phi ^\prime (z_n)\rightarrow 0~~\hbox {and}~~\Phi (z_n)\rightarrow c. \end{aligned}$$

(2.10)

Suppose to the contrary that $\Vert z_n\Vert \rightarrow \infty$ as $n\rightarrow \infty$. Setting $w_{n}=z_{n}/\Vert z_{n}\Vert$, then $\Vert w_{n}\Vert =1$. After passing to a subsequence, we may assume that $w_{n}\rightharpoonup w$ in E, $w_{n}\rightarrow w$ in $L_{loc}^{p}$ for $p\ge 1$, and $w_{n}(t)\rightarrow w(t)$ a.e. on $\mathbb {R}$. Let

$$\begin{aligned} \delta :=\lim _{n \rightarrow \infty }\sup \limits _{y\in {{\mathbb {R}}}} \int _{B(y,1)}|w_{n}^{+}|^{2}. \end{aligned}$$

If $\delta =0$, by the vanishing lemma (see [15, 29]), then $w^{+}_n\rightarrow 0$ in $L^{p}$ for any $p>2$. It follows from (2.2) that for any $s>0$,

$$\begin{aligned} \int _{\mathbb {R}}G(t,s|w_{n}^{+}|)\rightarrow 0. \end{aligned}$$

(2.11)

By virtue of (2.4) and (2.10), we have

$$\begin{aligned} \begin{aligned} 1+c\ge \Phi (z_{n})&\ge \Phi (r_{n}z_{n}+(-\,t_{n}z^{-}_{n})) +\Phi '(z_{n})\left( \frac{r^{2}_{n}-1}{2}z_{n}-r^{2}_{n}z_{n}^{-}\right) \\&=\frac{r_{n}^{2}}{2}\Vert z_{n}^{+}\Vert ^{2}-\int _{\mathbb {R}}G(t,r_{n}|z_{n}^{+}|)+o(1). \end{aligned} \end{aligned}$$

Let $r_{n}=s/\Vert z_{n}\Vert$, then by (2.11) we get

$$\begin{aligned} 1+c\ge \frac{s^{2}}{2}\Vert w_{n}^{+}\Vert ^{2}+o(1). \end{aligned}$$

(2.12)

Observe that by (2.3) we obtain

$$\begin{aligned} \Phi ^\prime (z_n)z_{n}\le \Vert z^{+}_{n}\Vert ^{2}-\Vert z^{-}_{n}\Vert ^{2}, \end{aligned}$$

and hence,

$$\begin{aligned} 2\Vert z^{+}_{n}\Vert ^{2}\ge \Vert z^{+}_{n}\Vert ^{2} +\Vert z^{-}_{n}\Vert ^{2}+\Phi ^\prime (z_n)z_{n}=\Vert z_{n}\Vert ^{2}+\Phi ^\prime (z_n)z_{n}, \end{aligned}$$

which implies that $\Vert w_{n}^{+}\Vert ^{2}\ge c_{0}$ for some $c_{0}>0$. Hence, (2.12) yields a contradiction if s is large enough. Then $\delta >0$. Going if necessary to a subsequence, we may assume the existence of $\{k_{n}\}\subset \mathbb {Z}$ such that

$$\begin{aligned} \int _{B(k_{n},2)}|w_{n}^{+}|^{2}>\frac{\delta }{2}. \end{aligned}$$

Let us define $\tilde{w}_{n}(x)=w_{n}(x+k_{n})$, then

$$\begin{aligned} \int _{B(0,2)}|\tilde{w}^{+}_{n}|^{2}>\frac{\delta }{2}. \end{aligned}$$

Therefore, passing to a subsequence, $\tilde{w}^{+}_{n}\rightarrow \tilde{w}^{+}$ in $L^{2}_{loc}$ and $\tilde{w}^{+}\ne 0$. Note that if $\tilde{w}\ne 0$, then $|z_{n}(t+k_n)|=|\tilde{w}_{n}(t)|\Vert z_{n}\Vert \rightarrow \infty$. Hence, it follows from $(g_{3})$ and Fatou’s lemma that

$$\begin{aligned} \begin{aligned} 0&=\lim _{n\rightarrow \infty }\frac{\Phi (z_{n})}{\Vert z_n\Vert ^{2}}\\&\le \lim _{n\rightarrow \infty }\left( \frac{1}{2}\left( \Vert w_{n}^{+}\Vert ^{2}-\Vert w_{n}^{-}\Vert ^{2}\right) -\int _{\mathbb {R}}\frac{G(t+k_n,|z_{n}(t+k_n)|)}{|z_{n}(t+k_n)|^{2}}|\tilde{w}_{n}|^{2}\right) \\&\rightarrow -\,\infty \end{aligned} \end{aligned}$$

since $\Vert w_{n}\Vert$ is bounded. Thus, we get a contradiction and the desired conclusion holds. $\square$

According to some arguments as [25, 32], we need to introduce a technical result.

Lemma 2.11

Let $(g_{1})$–$(g_{5})$ be satisfied, and assume that $\{z_{n}\}\subset E$ satisfies $z_{n}\rightharpoonup 0$ and $\varphi _{n}\in E$ is bounded. Then we have

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}}|g_{0}(t,|z_{n}|)-g(t,|z_{n}|)|z_{n}\varphi _{n}|\rightarrow 0,\\&\int _{\mathbb {R}}\left( G_{0}(t,|z_{n}|)-G(t,|z_{n}|)\right) \rightarrow 0. \end{aligned} \end{aligned}$$

Proof

For any $\varepsilon >0$, we define the set $U_{\varepsilon }(R)=\{t\in \mathbb {R}: |k(t)|\ge \varepsilon , |t|\ge R\}$. If $k\in \mathscr {K}$, we may find $R_{1}>0$ such that $|U_{\varepsilon }(R_{1})|< \varepsilon$. Since $z_{n}\rightharpoonup 0$, by Sobolev embedding we have for large n

$$\begin{aligned} \int _{B_{R_{1}}(0)}|z_n|^{p}<\varepsilon ~\hbox {for all}~p\ge 2. \end{aligned}$$

By $(g_{5})$–(ii), we get

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}}|g_{0}(t,|z_{n}|)z_{n}-g(t, |z_{n}|)z_{n}||\varphi _{n}|\le \int _{\mathbb {R}}|k(t)|(|z_n|+|z_n|^{q-1})|\varphi _{n}|\\&~~~=\left( \int _{U_{\varepsilon }(R_{1})}+\int _{B_{R_{1}}(0)}+\int _{\mathbb {R}\backslash (U_{\varepsilon }(R_{1})\cup B_{R_{1}}(0))}\right) |k(t)|(|z_n|+|z_n|^{q-1})|\varphi _{n}|\\&~~~:=\Pi _{1}+\Pi _{2}+\Pi _{3}. \end{aligned} \end{aligned}$$

Now we estimate each term the above appeared. In fact, since $k\in L^{\infty }$ and $q\ge 2$, by Hölder inequality we have

$$\begin{aligned} \begin{aligned}&\Pi _1\le |k|_{\infty }|U_{\varepsilon }(R_{1})|^{1/3}|z_n|_{3}|\varphi _{n}|_{3} +|k|_{\infty }|U_{\varepsilon }(R_{1})|^{(3-q)/3}|z_n|_{3}^{q-1}|\varphi _{n}|_{3}< c_{3}\varepsilon ,\\&\Pi _2\le |k|_{\infty }|\varphi _{n}|_{2}\left( \int _{B_{R_{1}}(0)}|z_n|^{2}\right) ^{1/2} +|h|_{\infty }|\varphi _{n}|_{q}\left( \int _{B_{R_{1}}(0)}|z_n|^{q}\right) ^{(q-1)/q}<c_{4}\varepsilon ,\\&\Pi _3\le \varepsilon (|z_n|_{2}|\varphi _{n}|_{2}+|z_n|_{q}^{q-1}|\varphi _{n}|_{q})< c_{5}\varepsilon . \end{aligned} \end{aligned}$$

Since $\varepsilon$ is arbitrary, we know that the first conclusion holds. Below we show that the second conclusion. Indeed, by the same way we can prove that

$$\begin{aligned} \int _{\mathbb {R}}\left( g_{0}(t,s|z_{n}|)-g(t,s|z_{n}|)\right) |z_{n}|^{2}\rightarrow 0,~\forall ~s\in [0,1]. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}}\left( G_{0}(t,|z_{n}|)-G(t,|z_{n}|)\right)&=\int _{\mathbb {R}}\left( \int _{0}^{1}(g_{0}(t,s|z_{n}|)-g(t,s|z_{n}|))s|z_n|^{2}ds\right) \\&=\int _{0}^{1}\left( \int _{\mathbb {R}}(g_{0}(t,s|z_{n}|)-g(t,s|z_{n}|))|z_n|^{2}\right) sds\\&\rightarrow 0. \end{aligned} \end{aligned}$$

The proof is completed. $\square$

In order to overcome the lack of compactness, we recall some known facts about the following limit problem

$$\begin{aligned} \dot{z}=\mathscr {J}H_{0z}(t,z) \end{aligned}$$

(2.13)

where $H_{0}(t,z)=\frac{1}{2}Lz\cdot z+G_{0}(t,z)$ with $G_{0z}(t,z)=g_{0}(t,|z|)z$, $g_{0}$ is 1-periodic in t and satisfies the conditions given in $(g_{5})$. We define the energy functional of limit problem (2.13)

$$\begin{aligned} \Phi _{0}(z)=\frac{1}{2}(\Vert z^+\Vert ^{2}-\Vert z^-\Vert ^{2})-\int _{\mathbb {R}}G_{0}(t,|z|), \end{aligned}$$

and the generalized Nehari manifold

$$\begin{aligned} \mathcal {M}_{0}:=\{z\in E\backslash E^{-}: \Phi _{0}'(z)z=0~\hbox {and}~\Phi _{0}'(z)w=0~\hbox {for any}~w\in E^{-}\}. \end{aligned}$$

According to [6], it is easy to see that problem (2.13) has a ground state homoclinic orbits $z_{0}$ satisfying

$$\begin{aligned} \Phi _{0}(z_{0})=m_{0}=\inf _{\mathcal {M}_{0}}\Phi _{0}>0. \end{aligned}$$

Next we use Lemma 2.11 to establish a global compactness result for bounded $(C)_{c}$-sequences.

Lemma 2.12

Let $\{z_{n}\}$ be a bounded $(C)_{c}$-sequences of $\Phi$ at level $c\ge 0$. Then there exist $z\in E$ such that $\Phi '(z)=0$, and there exist a number $k\in \mathbb {N}\cup \{0\}$, nontrivial critical points $z_{1},\ldots ,z_{k}$ of $\Phi _{0}$ and k sequences of points $\{t_{n}^{i}\}\subset \mathbb {Z}$, $1\le i\le k$, such that

$$\begin{aligned}&|t_{n}^{i}|\rightarrow +\infty , ~~|t_{n}^{i}-t_{n}^{j}|\rightarrow +\infty , ~if~i\ne j,~i,j=1,2,\ldots ,k, \\&\left\| z_{n}-z-\sum _{i=1}^{k}z_{i}(\cdot -t_{n}^{i})\right\| \rightarrow 0, \\&c=\Phi (z)+\sum _{i=1}^{k}\Phi _{0}(z_{i}). \end{aligned}$$

Proof

Let $\{z_{n}\}$ be a bounded $(C)_{c}$-sequences of $\Phi$ at level $c\ge 0$. Then, passing to a subsequence, we may assume that

$$\begin{aligned} z_{n}\rightharpoonup z~\hbox {in}~E,~z_{n}\rightarrow z~\hbox {in}~L_{loc}^{2},~z_{n}(t)\rightarrow z(t)~\hbox {a.e. on}~\mathbb {R}. \end{aligned}$$

It follows from Lemma 2.2 that $\Phi ^\prime (z)=0$. Denote $w_{n}=z_{n}-z$. Then $w_{n}\rightharpoonup 0$ in E, $w^{+}_{n}\rightharpoonup 0$ in $E^{+}$ and $w^{-}_{n}\rightharpoonup 0$ in $E^{-}$. Hence we can obtain

$$\begin{aligned} \begin{aligned}&\Vert w_{n}^{+}\Vert ^{2}=\Vert z_{n}^{+}\Vert ^{2}-\Vert z^{+}\Vert ^{2}+o(1),\\&\Vert w_{n}^{-}\Vert ^{2}=\Vert z_{n}^{-}\Vert ^{2}-\Vert z^{-}\Vert ^{2}+o(1),\\&\Vert w_{n}\Vert ^{2}=\Vert z_{n}\Vert ^{2}-\Vert z\Vert ^{2}+o(1). \end{aligned} \end{aligned}$$

(2.14)

Using Brezis–Lieb lemma, we get

$$\begin{aligned} \int _{\mathbb {R}}G(t,|w_{n}|)=\int _{\mathbb {R}}G(t,|z_{n}|)-\int _{\mathbb {R}}G(t,|z|)+o(1). \end{aligned}$$

(2.15)

Then by (2.14) and (2.15), we have

$$\begin{aligned} \Phi (w_n)=\Phi (z_n)-\Phi (z)+o(1). \end{aligned}$$

(2.16)

Moreover, by some similar arguments (see also [5]) we can get

$$\begin{aligned} \Phi ^\prime (w_n)=\Phi ^\prime (z_n)-\Phi ^\prime (z)+o(1). \end{aligned}$$

(2.17)

Now we distinguish two cases: $\{w_{n}\}$ is vanishing or $\{w_{n}\}$ is nonvanishing. If $\{w_{n}\}$ is vanishing, then

$$\begin{aligned} \lim _{n \rightarrow \infty }\sup \limits _{y\in {{\mathbb {R}}}} \int _{B(y,1)}|w_{n}|^{2}=0. \end{aligned}$$

In view of the vanishing lemma (see [15, 29]), then $w_n\rightarrow 0$ in $L^{p}$ for any $p>2$. Since the orthogonal projection of E on $E^{+}$ and $E^{-}$ is continuous in $L^{p}$, then $w_{n}^{+}\rightarrow 0$ and $w_{n}^{-}\rightarrow 0$ in $L^{p}$ for any $p>2$. Thus, it follows from (2.2) that

$$\begin{aligned} \int _{\mathbb {R}}g(t,|w_{n}|)w_{n}\cdot (w_{n}^{+}-w_{n}^{-})=o(1). \end{aligned}$$

(2.18)

Observe that $\{z_{n}\}$ is bounded $(C)_{c}$-sequences, then $\Phi ^\prime (z_n)=o(1)$, and it follows from (2.17) that $\Phi ^\prime (w_n)=o(1)$. Using this fact, we obtain

$$\begin{aligned} \begin{aligned} o(1)&=\Phi ^\prime (w_n)(w_{n}^{+}-w_{n}^{-})\\&=\Vert w_n\Vert ^{2} -\int _{\mathbb {R}}g(t,|w_{n}|)w_{n}\cdot (w_{n}^{+}-w_{n}^{-}). \end{aligned} \end{aligned}$$

Therefore, from (2.18), we deduce that $\Vert w_{n}\Vert \rightarrow 0$ in E, and so $z_{n}\rightarrow z$ in E and $k=0$.

If $\{w_{n}\}$ is nonvanishing, then there exist $\delta >0$, $\varrho >1$ and $\{y_{n}\}\subset \mathbb {Z}$ such that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\int _{B(y_n,\varrho )}|w_n|^{2}\ge \delta . \end{aligned}$$

(2.19)

It is clear that $\{y_n\}$ is unbounded. Passing to a subsequence, we may assume that $|y_n|\rightarrow \infty$. Let $\tilde{w}_{n}=w_n(t+y_n)$ and note that by (2.19) we find $z_{1}\ne 0$ such that up to a subsequence, $\tilde{w}_{n}\rightharpoonup z_{1}$ in E, $\tilde{w}_{n}\rightarrow z_{1}$ in $L_{loc}^{p}$ for $p\ge 2$ and $\tilde{w}_{n}(t)\rightarrow z_{1}(t)$ a.e. on $\mathbb {R}$. Next we claim that $z_{1}$ is a nontrivial critical point of $\Phi _{0}$. Indeed, for any $\varphi \in E$ and denote $\varphi _{n}=\varphi (\cdot -y_n)$, by Lemma 2.11 and the periodicity of $g_{0}$ we have

$$\begin{aligned} \begin{aligned} o(1)&=\Phi ^\prime (w_n)\varphi _{n}\\&=(w_n^+,\varphi _{n}^+)-(w_n^-,\varphi _{n}^-) -\int _{\mathbb {R}}g(t,|w_n|)w_{n}\cdot \varphi _{n}\\&=(w_n^+,\varphi _{n}^+)-(w_n^-,\varphi _{n}^-) -\int _{\mathbb {R}}g_{0}(t,|w_n|)w_{n}\cdot \varphi _{n}+o(1)\\&=(\tilde{w}_n^+,\varphi ^+)-(\tilde{w}_n^-,\varphi ^-) -\int _{\mathbb {R}}g_{0}(t,|\tilde{w}_n|)\tilde{w}_{n}\cdot \varphi +o(1)\\&=\Phi ^\prime _{0}(\tilde{w}_n)\varphi +o(1), \end{aligned} \end{aligned}$$

which implies that $\Phi ^\prime _{0}(z_{1})\varphi =0$ by Lemma 2.2 and $z_{1}$ is a nontrivial critical point of $\Phi _{0}$. Now denote $w_{n}^{1}=z_{n}-z-z_{1}(\cdot -y_n)$. Then by direct computation we have

$$\begin{aligned} \begin{aligned}&\Vert w_{n}^{1+}\Vert ^{2}=\Vert z_{n}^{+}\Vert ^{2}-\Vert z^{+}\Vert ^{2}-\Vert z_{1}^{+}\Vert ^{2}+o(1),\\&\Vert w_{n}^{1-}\Vert ^{2}=\Vert z_{n}^{-}\Vert ^{2}-\Vert z^{-}\Vert ^{2}-\Vert z_{1}^{-}\Vert ^{2}+o(1),\\&\Vert w_{n}^{1}\Vert ^{2}=\Vert z_{n}\Vert ^{2}-\Vert z\Vert ^{2}-\Vert z_{1}\Vert ^{2}+o(1). \end{aligned} \end{aligned}$$

(2.20)

Similar to (2.15), we get

$$\begin{aligned} \int _{\mathbb {R}}G(t,|w_{n}^{1}|)=\int _{\mathbb {R}}G(t,|z_{n}|) -\int _{\mathbb {R}}G(t,|z|)-\int _{\mathbb {R}}G(t,|z_{1}|)+o(1). \end{aligned}$$

(2.21)

From (2.20) and (2.21), we have

$$\begin{aligned} \Phi (w_n^{1})=\Phi (z_n)-\Phi (z)-\Phi _{0}(z_1)+o(1) \end{aligned}$$

(2.22)

and we take $t_{n}^{1}:=y_n$. Now we replace $w_{n}$ by $w_{n}^{1}$ and repeat the above argument in vanishing case and nonvanishing case, that is, if

$$\begin{aligned} \lim _{n \rightarrow \infty }\sup \limits _{y\in {{\mathbb {R}}}} \int _{B(y,1)}|w_{n}^{1}|^{2}=0, \end{aligned}$$

then $w_{n}^{1}\rightarrow 0$ in E and by (2.20) and (2.22) we take $k=1$. Otherwise as in nonvanishing case we find $\{y_{n}\}\subset \mathbb {Z}$ such that (2.19) holds for $\{w_{n}^{1}\}$. Then passing to a subsequence $|y_n|\rightarrow \infty$ and $|y_n-t_{n}^{1}|\rightarrow \infty$ as $n\rightarrow \infty$. Similar to the above argument, let $\tilde{w}_{n}^{1}(t)=w_n(t+y_n)$, then we can find $z_2\ne 0$ such that up to a subsequence, $\tilde{w}_{n}^{1}\rightharpoonup z_{2}$ in E, $\tilde{w}_{n}^{1}\rightarrow z_{2}$ in $L_{loc}^{p}$ for $p\ge 2$ and $\tilde{w}_{n}^{1}(t)\rightarrow z_{2}(t)$ a.e. on $\mathbb {R}$. Moreover, $z_2$ is a nontrivial critical point of $\Phi _{0}$ by Lemma 2.2. Denote $w_{n}^{2}=z_n-z-z_1(\cdot -t_{n}^{1})-z_2(\cdot -y_n)$, and similar to (2.20) and (2.22), we obtain

$$\begin{aligned} \begin{aligned}&\Vert w_{n}^{2+}\Vert ^{2}=\Vert z_{n}^{+}\Vert ^{2}-\Vert z^{+}\Vert ^{2}-\Vert z_{1}^{+}\Vert ^{2}-\Vert z_{2}^{+}\Vert ^{2}+o(1),\\&\Vert w_{n}^{2-}\Vert ^{2}=\Vert z_{n}^{-}\Vert ^{2}-\Vert z^{-}\Vert ^{2}-\Vert z_{1}^{-}\Vert ^{2}-\Vert z_{2}^{-}\Vert ^{2}+o(1),\\&\Vert w_{n}^{2}\Vert ^{2}=\Vert z_{n}\Vert ^{2}-\Vert z\Vert ^{2}-\Vert z_{1}\Vert ^{2}-\Vert z_{2}\Vert ^{2}+o(1),\\&\Phi (w_n^{2})=\Phi (z_n)-\Phi (z)-\Phi _{0}(z_1)-\Phi _{0}(z_2)+o(1), \end{aligned} \end{aligned}$$

and $t_{n}^{2}:=y_n$. Again we repeat the above arguments in vanishing case and nonvanishing case and the iterations must stop after finite steps, since there is a constant $\rho _{0}>0$ such that

$$\begin{aligned} \Vert z_{0}\Vert \ge \rho _{0}~\hbox {for any}~z_{0}\ne 0~\hbox {with}~\Phi ^\prime _{0}(z_{0})=0. \end{aligned}$$

(2.23)

In fact, in view of $\Phi ^\prime _{0}(z_{0})z_{0}^{\pm }=0$ and (2.2) we obtain

$$\begin{aligned} \Vert z_{0}^{+}\Vert ^{2}\le \int _{\mathbb {R}}|g_{0}(t,|z_{0}|)z_{0}z_{0}^{+}| \le \epsilon \Vert z_{0}^{+}\Vert \Vert z_{0}\Vert +C_{\epsilon }\Vert z_{0}^{+}\Vert \Vert z_{0}\Vert ^{p-1} \end{aligned}$$

and

$$\begin{aligned} \Vert z_{0}^{-}\Vert ^{2}\le \int _{\mathbb {R}}|g_{0}(t,|z_{0}|)z_{0}z_{0}^{-}| \le \epsilon \Vert z_{0}^{-}\Vert \Vert z_{0}\Vert +C_{\epsilon }\Vert z_{0}^{-}\Vert \Vert z_{0}\Vert ^{p-1}. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert z_{0}\Vert ^{2}\le 2\epsilon \Vert z_{0}\Vert ^{2}+2C_{\epsilon }\Vert z_{0}\Vert ^{p}, \end{aligned}$$

and (2.23) holds. The proof is completed. $\square$

3 The Proof of Theorem 1.1

In this section, we give the proof of the main results. Let $\mathcal {K}:=\{z\in E\backslash \{0\}: \Phi '(z)=0\}$ denote the set of all critical points of $\Phi$. To describe some properties of ground state homoclinic orbits, by using the standard bootstrap argument (see, e.g., [12] for the iterative steps) we can obtain the following regularity result (see also Lemma 2.3 in [10]).

Lemma 3.1

If $z\in \mathcal {K}$ with $|\Phi (z)|\le C_{1}$ and $|z|_{2}\le C_{2}$ , then, for any $q\in [2,+\,\infty )$, $z\in W^{1,q}(\mathbb {R},\mathbb {R}^{2N})$ with $\Vert z\Vert _{W^{1,q}}\le C_{q}$, where $C_{q}$ depends only on $C_{1}, C_{2}$ and q.

Let $\mathscr {L}$ be the set of all ground state homoclinic orbits of system (1.1). If $z\in \mathscr {L}$ then $\Phi (z)=m$, a standard argument shows that $\mathscr {L}$ is bounded in E; hence, $|z|_{2}\le C_{2}$ for all $z\in \mathscr {L}$ and some $C_{2}>0$. Therefore, as a consequence of Lemma 3.1 we see that, for each $q\in [2,+\,\infty )$, there is $C_{q}$ such that

$$\begin{aligned} \Vert z\Vert _{W^{1,q}}\le C_{q}~\hbox {for all}~z\in \mathscr {L}. \end{aligned}$$

This, together with the Sobolev embedding theorem, implies that there is $C_{\infty }>0$ with

$$\begin{aligned} |z|_{\infty }\le C_{\infty }~\hbox {for all}~z\in \mathscr {L}. \end{aligned}$$

(3.1)

Proof of Theorem 1.1

The proof will be carried out in several steps.

Step 1. Existence of ground state homoclinic orbits. Applying Lemma 2.8, we deduce that there exists a $(C)_{\tilde{c}}$-sequence $\{z_n\}$ of $\Phi$ such that

$$\begin{aligned} \Phi (z_n)\rightarrow \tilde{c}\le m~~\hbox {and}~~ \Vert \Phi '(z_n)\Vert (1+\Vert z_n\Vert )\rightarrow 0. \end{aligned}$$

It follows from Lemma 2.10 that $\{z_{n}\}$ is bounded; then, passing to a subsequence, $z_{n}\rightharpoonup z$ in E, $z_{n}\rightarrow z$ in $L_{loc}^{p}$ for $p\ge 2$ and $z_{n}(t)\rightarrow z(t)$ a.e. on $\mathbb {R}$, and $\Phi ^\prime (z)=0$. If $z\ne 0$, then z is a nontrivial critical point of $\Phi$. By (2.3) and Fatou’s lemma, we have

$$\begin{aligned} m\ge & {} \tilde{c}=\lim _{n\rightarrow \infty }\left( \Phi (z_n)-\frac{1}{2}\Phi '(z_n)z_n\right) \\= & {} \lim _{n\rightarrow \infty }\int _{\mathbb {R}}\left( \frac{1}{2}g(t,|z_n|)|z_n|^{2}-G(t,|z_n|)\right) \\\ge & {} \int _{\mathbb {R}}\lim _{n\rightarrow \infty }\left( \frac{1}{2}g(t,|z_n|)|z_n|^{2}-G(t,|z_n|)\right) \\= & {} \Phi (z)-\frac{1}{2}\Phi '(z)z=\Phi (z), \end{aligned}$$

which implies that $\Phi (z)\le m$. Hence, $\Phi (z)= m=\inf _{z\in \mathcal {M}}\Phi$ and z is a ground state homoclinic orbit of system (1.1).

Next we claim that $z\ne 0$. Indeed, observe that the limit problem (2.13) has a ground state solution $z_{0}\in \mathcal {M}_{0}$ such that $\Phi _{0}(z_0)= m_{0}$. It follows from Lemma 2.9 that there exist $t_{0}>0$ and $w_{0}\in E^{-}$ such that $t_0z_0+w_0\in \mathcal {M}$, and $\Phi (t_0z_0+w_0)\ge m$. Therefore, by $(g_{5})$-(i) and Lemma 2.4, we have

$$\begin{aligned} m_{0}=\Phi _{0}(z_0)\ge \Phi _{0}(t_0z_0+w_0)>\Phi (t_0z_0+w_0)\ge m\ge \tilde{c}, \end{aligned}$$

then by Lemma 2.12 we get $k=0$ and $z_n\rightarrow z$ in E, and so $z\ne 0$. The proof is completed.

Step 2: $\mathscr {L}$ is compact in $H^{1}(\mathbb {R},\mathbb {R}^{2N})$ (Compactness). It follows from Step 1 that $\mathscr {L}\ne \emptyset$. Let $\{z_{n}\}\subset \mathscr {L}$, then $z_{n}\in \mathcal {M}$, $\Phi (z_n)=m$ and $\Phi '(z_{n})=0$. Thus, $\{z_{n}\}$ is a $(C)_{m}$-sequence. By Lemma 2.10, $\{z_{n}\}$ is bounded. Similar to the proof of Step 1, passing to a subsequence, we can deduce that there exists z such that $z_{n}\rightarrow z$ in E and $z\in \mathscr {L}$. Observe that $Az=g(t,|z|)z$. Since $z_{n}$ and z are solutions of system (1.1), we have

$$\begin{aligned} A(z_n-z)=g(t,|z_{n}|)z_{n}-g(t,|z|)z, \end{aligned}$$

which implies that

$$\begin{aligned} \begin{aligned} |A(z_{n}-z)|_{2}&=|g(t,|z_{n}|)z_{n}-g(t,|z|)z|_{2}\\&\le |g(t,|z_{n}|)(z_{n}-z)|_{2}+|(g(t,|z_{n}|)-g(t,|z|))z|_{2}. \end{aligned} \end{aligned}$$

By (3.1) and the facts that $z_{n}\rightarrow z$ in E and the decay of integral of z, there holds

$$\begin{aligned} \int _{\mathbb {R}}|g(t,|z_{n}|)|^{2}|z_{n}-z|^{2}\le c_{2}\int _{\mathbb {R}}|z_{n}-z|^{2}\rightarrow 0 \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {R}}|(g(t,|z_{n}|)-g(t,|z|))z|^{2}=\left( \int _{|t|\le R}+\int _{|t|\ge R}\right) |(g(t,|z_{n}|)-g(t,|z|))z|^{2}\rightarrow 0. \end{aligned}$$

Therefore, we get $|A(z_{n}-z)|_{2}=o(1)$, which implies that $z_{n}\rightarrow z$ in $H^{1}(\mathbb {R},\mathbb {R}^{2N})$.

Step 3. Exponential decay of ground state homoclinic orbits. Observe that

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}z=\mathscr {J}(Lz+g(t,|z|)z) \end{aligned}$$

and

$$\begin{aligned} \frac{\hbox {d}^{2}}{\hbox {d}t^{2}}z=(\mathscr {J}L)^{2}z+W(t,z) \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} W(t,z)&=\mathscr {J}\left( g_{t}(t,|z|)z+L\mathscr {J}g(t,|z|)z+(g_{z}(t,|z|)z+g(t,|z|))\mathscr {J}Lz\right. \quad \\&~~~\left. +\,(g_{z}(t,|z|)z+g(t,|z|))\mathscr {J}g(t,|z|)z\right) . \end{aligned} \end{aligned}$$

(3.2)

Letting

$$\begin{aligned} \hbox {sgn}z=\left\{ \begin{array}{ll} \frac{z}{|z|}&{}\quad \hbox {if}\;z\ne 0,\\ 0&{}\quad \hbox {if}\;z=0. \end{array} \right. \end{aligned}$$

By the Kato’s inequality and (3.2), using the real positivity of $(\mathscr {J}L)^{2}$ we get

$$\begin{aligned} \frac{\hbox {d}^{2}}{\hbox {d}t^{2}}|z|\ge \frac{\hbox {d}^{2}}{\hbox {d}t^{2}}z(\hbox {sgn}z) =(\mathscr {J}L)^{2}z\frac{z}{|z|}+W(t,z)\frac{z}{|z|}\ge \nu |z|-|W(t,z)| \end{aligned}$$

(3.3)

for some $\nu >0$. Hence by (2.1), (3.1) and (3.3), there exists $\Lambda >0$ such that

$$\begin{aligned} \frac{\hbox {d}^{2}}{\hbox {d}t^{2}}|z|\ge -\Lambda |z|~\hbox {for all}~t\in \mathbb {R}. \end{aligned}$$

It then follows from the sub-solution estimate [21] that

$$\begin{aligned} |z(t)|\le C_{0}\int _{B_{1}(t)}|z(s)|\hbox {d}s \end{aligned}$$

(3.4)

with $C_{0}$ independent of t and $z\in \mathscr {L}$. Since $\mathscr {L}$ is compact in $H^{1}(\mathbb {R},\mathbb {R}^{2N})$, then $|z(t)|\rightarrow 0$ as $|t|\rightarrow \infty$ uniformly in $z\in \mathscr {L}$. In fact, if not, then by (3.4) there exist $c_{0}>0$, $z_{j}\in \mathscr {L}$ and $t_{j}\in \mathbb {R}$ with $|t_{j}|\rightarrow \infty$ such that

$$\begin{aligned} c_{0}\le |z_{j}(t_{j})|\le C_{0}\int _{B_{1}(t_{j})}|z_{j}|. \end{aligned}$$

We may assume that $z_{j}\rightarrow z\in \mathscr {L}$ in $H^{1}(\mathbb {R},\mathbb {R}^{2N})$, by the compactness of $\mathscr {L}$, then we get

$$\begin{aligned} \begin{aligned} c_{0}&\le |z_{j}(t_{j})|\le C_{0}\int _{B_{1}(t_{j})}|z_{j}|\le C_{0}\int _{B_{1}(t_{j})}|z_{j}-z|+ C_{0}\int _{B_{1}(t_{j})}|z|\\&\le \tilde{C}\left( \int _{\mathbb {R}}|z_{j}-z|^{2}\right) ^{1/2} +C_{0}\int _{B_{1}(t_{j})}|z|\rightarrow 0, \end{aligned} \end{aligned}$$

which implies a contradiction. Note that $g(t,s)=o(1)$, $g_{t}(t,s)=o(1)$ and $g_{s}(t,s)s=o(1)$ as $s\rightarrow 0$; hence, we can choose $0<\delta <\frac{\nu }{2}$ and $R>0$ such that $|W(t,z)|\le \frac{\nu }{2}|z|$ for $|t|\ge R$. This, together with (3.3), implies that

$$\begin{aligned} \frac{\hbox {d}^{2}}{\hbox {d}t^{2}}|z|\ge \delta |z|,~\hbox {for all}~|t|\ge R, z\in \mathscr {L}. \end{aligned}$$

Let $\Gamma (t)$ be a fundamental solution to $-\frac{\hbox {d}^{2}}{\hbox {d}t^{2}}\Gamma +\delta \Gamma =0$. Using the uniform boundedness, we may choose $\Gamma (t)$ so that $|z(t)|\le \delta \Gamma (t)$ holds on $|t|=R$ for all $z\in \mathscr {L}$. Let $w=|z|-\delta \Gamma$, then

$$\begin{aligned} \frac{\hbox {d}^{2}}{\hbox {d}t^{2}} w=\frac{\hbox {d}^{2}}{\hbox {d}t^{2}}|z|-\delta \frac{\hbox {d}^{2}}{\hbox {d}t^{2}}\Gamma \ge \delta (|z|-\delta \Gamma )=\delta w,~\hbox {for}~|t|\ge R. \end{aligned}$$

By the maximum principle, we can conclude that $w(t)\le 0$ for $|t|\ge R$, i.e., $|z(t)|\le \delta \Gamma (t)$ for $|t|\ge R$. It is well known that there exist $C'>0$ such that

$$\begin{aligned} \Gamma (t)\le C' \exp \left( -\,\sqrt{\delta }|t|\right) \end{aligned}$$

for $|t|\ge 1$. Hence, we get

$$\begin{aligned} |z(t)|\le C \exp \left( -\,c|t|\right) \end{aligned}$$

for $t\in \mathbb {R}$ and some $C, c>0$. The proof is completed. $\square$