Homoclinic orbits for second order Hamiltonian systems with asymptotically linear terms at infinity

Abstract

In this paper, by using some different asymptotically linear conditions from those previously used in Hamiltonian systems, we obtain the existence of nontrivial homoclinic orbits for a class of second order Hamiltonian systems by the variational method.

MSC:37J45, 37K05, 58E05.

1 Introduction and main result

We consider the following second order Hamiltonian system:

$$u^{″}(t)-A(t)u(t)+\mathrm{\nabla }H(t,u(t))=0,t\in \mathbb{R},$$

(1.1)

where $H\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$ is T-periodic in t, $\mathrm{\nabla }H(t,x)$ denotes its gradient with respect to the x variable, and $A(t)$ is the T-periodic $N\times N$ matrix that satisfies

$$A(t)\in C(\mathbb{R},{\mathbb{R}}^{N^2}),$$

(1.2)

and it is symmetric and positive definite uniformly for $t\in [0,T]$. We say that a solution $u(t)$ of (1.1) is homoclinic (with 0) if $u(t)\in C^2(\mathbb{R},{\mathbb{R}}^N)$ such that $u(t)\to 0$ and $u^{\prime }(t)\to 0$ as $|t|\to \mathrm{\infty }$. If $u(t)\not\equiv 0$, then $u(t)$ is called a nontrivial homoclinic solution.

Let $G(t,u):=\frac{1}{2}(\mathrm{\nabla }H(t,u),u)-H(t,u)$. We assume:

(H₁) $H\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$ is T-periodic in t, and $H(t,u)\ge 0$, $\mathrm{\forall }(t,u)\in \mathbb{R}\times {\mathbb{R}}^N$.

(H₂) There are some constants $c,R_1>0$ and $\mu >2$ such that $|\mathrm{\nabla }H(t,u)|\le c{|u|}^{\mu -1}$ if $|u|\le R_1$.

(H₃) There is a constant $V>0$ such that

$$H(t,u)=\frac{1}{2}V{|u|}^2+F(t,u),|\mathrm{\nabla }F(t,u)|=o(|u|)\text{as }|u|\to \mathrm{\infty }.$$

(H₄) $G(t,u)\ge 0$, $\mathrm{\forall }(t,u)\in \mathbb{R}\times {\mathbb{R}}^N$, and there exist $\alpha \in (1,2)$, $c_1,c_2>0$, and $R_2>R_1$ such that

$$G(t,u)\ge c_1{|u|}^{\mu }\text{if }|u|\le R_1,G(t,u)\ge c_2{|u|}^{\alpha }\text{if }|u|\ge R_2.$$

Let

$$a_0:=\underset{t\in \mathbb{R}}{max}\underset{u\in {\mathbb{R}}^N,|u|=1}{sup}(A(t)u,u).$$

Now, our main result reads as follows.

Theorem 1.1 If (1.2) and (H₁)-(H₄) with $V>a_0$ hold, then (1.1) has a nontrivial homoclinic orbit.

Example 1.1 Let

$$H(t,u)=\{\begin{array}{ll}(\frac{1}{2}V-h(t)){|u|}^{\mu }& \text{if }|u|\le 1,\\ \frac{1}{2}V{|u|}^2-h(t){|u|}^{\alpha }& \text{if }|u|\ge 1,\end{array}$$

where $h\in C^1(\mathbb{R},\mathbb{R})$ is T-periodic in t, $0<{inf}_{t\in \mathbb{R}}h(t)\le {sup}_{t\in \mathbb{R}}h(t)<\frac{1}{2}V$ and $\mu >2>\alpha >1$. It is not hard to check that the above function satisfies (H₁)-(H₄).

We will use the following theorem to prove our main result.

Theorem A ([1])

Let E be a Banach space equipped with the norm $\parallel \cdot \parallel$ and let $J\subset {\mathbb{R}}^{+}$ be an interval. We consider a family ${(I_{\lambda })}_{\lambda \in J}$ of $C^1$-functionals on E of the form

$$I_{\lambda }(u)=A(u)-\lambda B(u),$$

where $B(u)\ge 0$, $\mathrm{\forall }u\in E$ and such that either $A(u)\to +\mathrm{\infty }$ or $B(u)\to +\mathrm{\infty }$ as $\parallel u\parallel \to +\mathrm{\infty }$. We assume there are two points $(v_1,v_2)$ in E such that setting

$$\mathrm{\Gamma }=\{\gamma \in C([0,1],E),\gamma (0)=v_1,\gamma (1)=v_2\}$$

we have, $\mathrm{\forall }\lambda \in J$,

$$c_{\lambda }:=\underset{\gamma \in \mathrm{\Gamma }}{inf}\underset{t\in [0,1]}{max}{I}_{\lambda }(\gamma (t))>max\{I_{\lambda }(v_1),I_{\lambda }(v_2)\}.$$

Then, for almost every $\lambda \in J$, there is a sequence $\{v_n\}\subset E$ such that

$$\{v_n\}\mathit{\text{ is bounded}},I_{\lambda }(v_n)\to c_{\lambda }\mathit{\text{and}}{I}_{\lambda }^{\prime }(v_n)\to 0\mathit{\text{in the dual }}{E}^{-1}\mathit{\text{ of }}E.$$

In recent decades, many authors are devoted to the existence and multiplicity of homoclinic orbits for second order Hamiltonian systems with super or asymptotically linear terms by critical point theory, see [2–22] and the references therein. Many authors [2–7, 9, 11–13, 15, 17–19] have studied the existence of homoclinic orbits of (1.1) by considering the following so-called global Ambrosetti-Rabinowitz condition on H due to Ambrosetti and Rabinowitz (e.g., [3]): there exists a constant $\mu >2$ such that

$$0<\mu H(t,u)\le (\mathrm{\nabla }H(t,u),u),u\in {\mathbb{R}}^N\mathrm{\setminus }\{0\},$$

(1.3)

where $(\cdot ,\cdot )$ denotes the inner product in ${\mathbb{R}}^N$, and the corresponding norm is denoted by $|\cdot |$. Roughly speaking the role of (1.3) is to insure that all Palais-Smale sequences for the corresponding function of (1.1) at the mountain-pass level are bounded. By removing or weakening the condition (1.3), some authors studied the homoclinic orbits of (1.1). For example, Zou and Li [22] proved that the system (1.1) has infinitely many homoclinic orbits by using the variant fountain theorem; Chen [8] obtained the existence of a ground state homoclinic orbit for (1.1) by a variant generalized weak linking theorem due to Schechter and Zou. Ou and Tang [16] obtained the existence of a homoclinic solution of (1.1) by the minimax methods in the critical point theory. For second order Hamiltonian systems without periodicity, we refer the readers to [20–22] and so on.

The rest of our paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proof of our result. In Section 3, we give the detailed proof of our result.

2 Preliminary lemmas

Throughout this paper we denote by ${\parallel \cdot \parallel }_{L^q}$ the usual $L^q(\mathbb{R},{\mathbb{R}}^N)$ norm and C for generic constants.

In what follows, we always assume (1.2) and (H₁)-(H₄) with $V>a_0$ hold. Let $E:=H^1(\mathbb{R},{\mathbb{R}}^N)$ under the usual norm

$${\parallel u\parallel }_E^2={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}({|u|}^2+|u^{\prime }{|}^2)dt.$$

Thus E is a Hilbert space and it is not difficult to show that $E\subset C^0(\mathbb{R},{\mathbb{R}}^N)$, the space of continuous functions u on ℝ such that $u(t)\to 0$ as $|t|\to \mathrm{\infty }$ (see, e.g., [18]). We will seek solutions of (1.1) as critical points of the functional I associated with (1.1) and given by

$$I(u):=\frac{1}{2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\left(\right|u^{\prime }{|}^2+(A(t)u,u))dt-{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}H(t,u)dt.$$

We define a new norm

$${\parallel u\parallel }^2:={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}((A(t)u,u)+|u^{\prime }{|}^2)dt,$$

and its corresponding inner product is denoted by $〈\cdot ,\cdot 〉$. By (1.2), $\parallel \cdot \parallel$ can and will be taken as an equivalent norm on E. Hence I can be written as

$$I(u):=\frac{1}{2}{\parallel u\parallel }^2-{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}H(t,u)dt.$$

(2.1)

The assumptions on H imply that $I\in C^1(E,\mathbb{R})$. Moreover, critical points of I are classical solutions of (1.1) satisfying $u^{\prime }(t)\to 0$ as $|t|\to \mathrm{\infty }$. Thus u is a homoclinic solution of (1.1). Let us show that I has a mountain-pass geometry. Since $I(0)=0$ this is a consequence of the two following results.

Lemma 2.1 $I(u)=\frac{1}{2}{\parallel u\parallel }^2+o({\parallel u\parallel }^2)$ as $u\to 0$.

Proof By (H₂) and (H₃), we know for any $\epsilon >0$ there exists a $C_{\epsilon }>0$ such that

$$|\mathrm{\nabla }H(t,u)|\le \epsilon |u|+C_{\epsilon }{|u|}^{p-1},\mathrm{\forall }(t,u)\in \mathbb{R}\times {\mathbb{R}}^N,$$

(2.2)

where $p>2$. It follows from $\frac{1}{2}(\mathrm{\nabla }H(t,u),u)\ge H(t,u)$ (see (H₄)) that

$$|H(t,u)|\le \frac{\epsilon }{2}{|u|}^2+\frac{C_{\epsilon }}{2}{|u|}^p,\mathrm{\forall }(t,u)\in \mathbb{R}\times {\mathbb{R}}^N.$$

(2.3)

By (2.3) and the Sobolev embedding theorem, we deduce that

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}|H(t,u)|dt\le \frac{\epsilon }{2}{\parallel u\parallel }^2+C{\parallel u\parallel }^p,$$

which implies the conclusion. □

Lemma 2.2 There is a function $v\in E$ with $v\ne 0$ satisfying $I(v)\le 0$.

Proof Let

$$\begin{array}{c}{d}^2:=N{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{e}^{-2\alpha t^2}dt,0<\alpha <V-a_0,\hfill \\ w_{\alpha ,i}(t):=\frac{1}{d}{e}^{-\alpha t^2},i=1,\dots ,N\text{and}{w}_{\alpha }(t):=(w_{\alpha ,1}(t),\dots ,w_{\alpha ,N}(t)).\hfill \end{array}$$

Obviously, $w_{\alpha ,i}^{\prime }(t):=-\frac{1}{d}2\alpha t{e}^{-\alpha t^2}$, $i=1,\dots ,N$. Straightforward calculations show that

$${\parallel w_{\alpha }\parallel }_{L^2}=1\text{and}{\parallel w_{\alpha }^{\prime }\parallel }_{L^2}^2=\alpha .$$

(2.4)

For every $t\in \mathbb{R}$, $|s{w}_{\alpha }|\to +\mathrm{\infty }$ as $s\to \mathrm{\infty }$. It follows from (H₃) that

$$\underset{s\to \mathrm{\infty }}{lim}\frac{H(t,s{w}_{\alpha })}{s^2}=\underset{s\to \mathrm{\infty }}{lim}\frac{H(t,s{w}_{\alpha })}{s^2{|w_{\alpha }|}^2}{|w_{\alpha }|}^2=\frac{1}{2}V{|w_{\alpha }|}^2,\text{a.e. }t\in \mathbb{R},$$

which together with (2.4), the definition of $a_0$ (above Theorem 1.1) and the Fatou lemma implies

$$\begin{array}{rcl}\underset{s\to \mathrm{\infty }}{lim}\frac{I(s{w}_{\alpha })}{s^2}& =& \frac{1}{2}{\parallel w_{\alpha }^{\prime }\parallel }_{L^2}^2+\frac{1}{2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}(A(t)w_{\alpha },w_{\alpha })dt-\underset{s\to \mathrm{\infty }}{lim}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{H(t,s{w}_{\alpha })}{s^2}dt\\ \le & \frac{1}{2}\alpha +\frac{a_0}{2}{\parallel w_{\alpha }\parallel }_{L^2}^2-\underset{s\to \mathrm{\infty }}{lim}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{H(t,s{w}_{\alpha })}{s^2}dt\\ =& \frac{1}{2}\alpha +\frac{a_0}{2}-\frac{1}{2}V<0.\end{array}$$

Therefore, we can choose $v:=s{w}_{\alpha }$ with s big enough such that $v\in E$ with $v\ne 0$ satisfying $I(v)\le 0$. □

We define on E the family of functionals

$$I_{\lambda }(u):=\frac{1}{2}{\parallel u\parallel }^2-\lambda {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}H(t,u)dt,\lambda \in [1,2].$$

(2.5)

Lemma 2.3 The family $(I_{\lambda })$ with $\lambda \in [1,2]$ satisfies the hypotheses of Theorem  A. In particular for almost every $\lambda \in [1,2]$ there is a bounded sequence $\{v_j\}\subset E$ satisfying

$$I_{\lambda }(v_j)\to c_{\lambda }\mathit{\text{and}}{I}_{\lambda }^{\prime }(v_j)\to 0.$$

Proof For the $v\in E$ obtained in Lemma 2.2, we have $I_{\lambda }(v)\le 0$. It follows from (H₁) that $I_{\lambda }(v)\le I(v)\le 0$, $\mathrm{\forall }\lambda \in [1,2]$. By the proof in Lemma 2.1, we have

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}H(t,u)dt=o\left({\parallel u\parallel }^2\right)\text{as }u\to 0.$$

(2.6)

Let

$$\mathrm{\Gamma }:=\{\gamma \in C([0,1],E):\gamma (0)=0\text{ and }\gamma (1)=v\},$$

then it follows from (2.5) and (2.6) that

$$c_{\lambda }:=\underset{\gamma \in \mathrm{\Gamma }}{inf}\underset{s\in [0,1]}{max}{I}_{\lambda }(\gamma (s))>0,\mathrm{\forall }\lambda \in [1,2].$$

An application of Theorem A now completes the proof. □

Lemma 2.4 If $\{u_j\}\subset E$ vanishes and is bounded, then

$$\underset{j\to \mathrm{\infty }}{lim}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}G(t,u_j)dt=0.$$

Proof It is known that if $\{u_j\}$ vanishes, then $u_j\to 0$ in $L^q(\mathbb{R},{\mathbb{R}}^N)$ for all $q\in (2,\mathrm{\infty })$, which together with (2.2), (2.3), and the Hölder inequality implies

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}(\mathrm{\nabla }H(t,u_j),u_j)\le \epsilon {\parallel u_j\parallel }_{L^2}^2+C_{\epsilon }{\parallel u_j\parallel }_{L^p}^p\to 0$$

and

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}H(t,u_j)dt\le \frac{\epsilon }{2}{\parallel u_j\parallel }_{L^2}^2+\frac{C_{\epsilon }}{2}{\parallel u_j\parallel }_{L^p}^p\to 0.$$

Therefore, the proof follows from the definition of G. □

Lemma 2.5 If $\{u_j\}$ is bounded in E and satisfies

$$0<\underset{j\to \mathrm{\infty }}{lim}{I}_{\lambda }(u_j)\le c_{\lambda }\mathit{\text{and}}{I}_{\lambda }^{\prime }(u_j)\to 0,$$

then up to a subsequence, $u_j\rightharpoonup u_{\lambda }\ne 0$ with $I_{\lambda }(u_{\lambda })\le c_{\lambda }$ and $I_{\lambda }^{\prime }(u_{\lambda })=0$.

Proof Note that $\{u_j\}$ is bounded and

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}G(t,u_j)dt=I_{\lambda }(u_j)-\frac{1}{2}{I}_{\lambda }^{\prime }(u_j)u_j\to \underset{j\to \mathrm{\infty }}{lim}{I}_{\lambda }(u_j)>0,$$

it follows from Lemma 2.4 that $\{u_j\}$ does not vanish, i.e., there are $r,\delta >0$ and a sequence $\{s_j\}\subset \mathbb{R}$ such that

$$\underset{j\to \mathrm{\infty }}{lim}{\int }_{B_r(s_j)}{u}_j^{2}dt\ge \delta ,$$

(2.7)

where $B_r(s_j):=[s_j-r,s_j+r]$. The fact that $\{u_j\}$ is bounded implies that $u_j\rightharpoonup u_{\lambda }$ in E and $u_j\to u_{\lambda }$ in $L_{\mathrm{loc}}^2(\mathbb{R},{\mathbb{R}}^N)$ (see [23]) after passing to a subsequence, thus we get $u_{\lambda }\ne 0$ by (2.7). By $I_{\lambda }^{\prime }(u_j)\to 0$ and the fact $I_{\lambda }^{\prime }$ is weakly sequentially continuous [24], we have

$$I_{\lambda }^{\prime }(u_{\lambda })v=\underset{j\to \mathrm{\infty }}{lim}{I}_{\lambda }^{\prime }(u_j)v=0,\mathrm{\forall }v\in E.$$

It implies that $I_{\lambda }^{\prime }(u_{\lambda })=0$.

Observe that (H₄) implies $G(t,u)\ge 0$ for all $(t,u)\in \mathbb{R}\times {\mathbb{R}}^N$, which together with the Fatou lemma and $I_{\lambda }^{\prime }(u_{\lambda })=0$ implies

$$\begin{array}{rcl}{c}_{\lambda }& \ge & \underset{j\to \mathrm{\infty }}{lim}(I_{\lambda }(u_j)-\frac{1}{2}{I}_{\lambda }^{\prime }(u_j)u_j)\\ =& \underset{j\to \mathrm{\infty }}{lim}\lambda {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}G(t,u_j)dt\\ \ge & \lambda {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}G(t,u_{\lambda })dt\\ =& I_{\lambda }(u_{\lambda })-\frac{1}{2}{I}_{\lambda }^{\prime }(u_{\lambda })u_{\lambda }=I_{\lambda }(u_{\lambda }).\end{array}$$

Therefore, the proof is finished. □

By Lemmas 2.3 and 2.5, we deduce the existence of a sequence $\{({\lambda }_j,u_j)\}\subset [1,2]\times E$ such that:

$$\begin{array}{r}•{\lambda }_j\to 1\text{ and }\{{\lambda }_j\}\text{ is decreasing},\\ •u_j\ne 0,I_{{\lambda }_j}(u_j)\le c_{{\lambda }_j}\text{ and }{I}_{{\lambda }_j}^{\prime }(u_j)=0.\end{array}$$

(2.8)

Since

$$\frac{1}{2}{\parallel u_j\parallel }^2-{\lambda }_j{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}H(t,u_j)dt\le c_{{\lambda }_j}\text{and}{\parallel u_j\parallel }^2={\lambda }_j{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}(\mathrm{\nabla }H(t,u_j),u_j)dt,$$

we have

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}G(t,u_j)dt\le \frac{c_{{\lambda }_j}}{{\lambda }_j}.$$

Clearly $\frac{c_{{\lambda }_j}}{{\lambda }_j}$ is increasing and bounded by $c=c_1$, and it follows that

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}G(t,u_j)dt\le c,\mathrm{\forall }j\in \mathbb{N}.$$

(2.9)

Lemma 2.6 The sequence $\{u_j\}$ obtained in (2.8) is bounded.

Proof Since $H\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$ is T-periodic in t, by (H₄) and (H₃), respectively,

$$\begin{array}{rcl}{\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}G(t,u_j)dt& =& {\int }_{\{t\in \mathbb{R}:R_1\le |u_j|\le R_2\}\cup \{t\in \mathbb{R}:|u_j|\ge R_2\}}G(t,u_j)dt\\ \ge & c_1^{\prime }{\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}{|u_j|}^{\alpha }dt\end{array}$$

(2.10)

and

$${\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}|\mathrm{\nabla }H(t,u_j)|\cdot |u_j|dt\le c_2^{\prime }{\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}{|u_j|}^{2}dt$$

(2.11)

for some positive constants $c_1^{\prime }$ and $c_2^{\prime }$. Note that (2.8) implies

$$\frac{I_{{\lambda }_j}(u_j)-\frac{1}{2}{I}_{{\lambda }_j}^{\prime }(u_j)u_j}{{\lambda }_j}\le C,$$

thus it follows from (H₄) and (2.10) that

$$\begin{array}{rcl}C& \ge & \frac{I_{{\lambda }_j}(u_j)-\frac{1}{2}{I}_{{\lambda }_j}^{\prime }(u_j)u_j}{{\lambda }_j}\\ =& {\int }_{\mathbb{R}}G(t,u_j)dt\\ =& {\int }_{\{t\in \mathbb{R}:|u_j|\le R_1\}}G(t,u_j)dt+{\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}G(t,u_j)dt\\ \ge & c_1{\int }_{\{t\in \mathbb{R}:|u_j|\le R_1\}}{|u_j|}^{\mu }dt+c_1^{\prime }{\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}{|u_j|}^{\alpha }dt.\end{array}$$

(2.12)

Take $s\in (0,\frac{\alpha }{2})$, then by (2.12), the Hölder inequality, and the Sobolev imbedding theorem,

$$\begin{array}{c}{\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}{|u_j|}^{2}dt\hfill \\ ={\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}{|u_j|}^{2s}{|u_j|}^{2(1-s)}dt\hfill \\ \le {\left({\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}{|u_j|}^{\alpha }\right)}^{\frac{2s}{\alpha }}{\left({\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}{|u_j|}^{\frac{2\alpha (1-s)}{\alpha -2s}}\right)}^{\frac{\alpha -2s}{\alpha }}\hfill \\ \le C_1{\parallel u_j\parallel }^{2(1-s)}\hfill \end{array}$$

(2.13)

for some positive constant $C_1$, where $\frac{2\alpha (1-s)}{\alpha -2s}\ge 2$. Note that $I_{{\lambda }_j}^{\prime }(u_j)u_j=0$, it follows from (H₂), (2.11)-(2.13), the Hölder inequality and the Sobolev imbedding theorem that

$$\begin{array}{rl}{\parallel u_j\parallel }^2& ={\lambda }_j{\int }_{\mathbb{R}}(\mathrm{\nabla }H(t,u_j),u_j)dt\\ \le C_2{\int }_{\{t\in \mathbb{R}:|u_j|\le R_1\}}|\mathrm{\nabla }H(t,u_j)|\cdot |u_j|dt+C_2{\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}|\mathrm{\nabla }H(t,u_j)|\cdot |u_j|dt\\ \le C_3{\int }_{\{t\in \mathbb{R}:|u_j|\le R_1\}}{|u_j|}^{\mu -1}\cdot |u_j|dt+C_3{\int }_{\{t\in \mathbb{R}:|u_j|\ge R_1\}}{|u_j|}^{2}dt\\ \le C_4+C_3{C}_1{\parallel u_j\parallel }^{2(1-s)}\end{array}$$

(2.14)

for some positive constants $C_2$, $C_3$, and $C_4$, where $0<2(1-s)<2$. Therefore, (2.14) implies that $\{u_j\}$ is bounded and the proof is finished. □

3 Proof of main result

We are now in a position to prove our main result.

Proof of Theorem 1.1 $\{u_j\}$ is bounded by Lemma 2.6, so we can assume $u_j\rightharpoonup u$ and $u_j\to u$ a.e. $t\in \mathbb{R}$, up to a subsequence. By (2.8), we have

$$\underset{j\to \mathrm{\infty }}{lim}{I}^{\prime }(u_j)\phi =\underset{j\to \mathrm{\infty }}{lim}(I_{{\lambda }_j}^{\prime }(u_j)\phi +({\lambda }_j-1){\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}(\mathrm{\nabla }H(t,u_j),\phi )dt)=0,\mathrm{\forall }\phi \in E.$$

Note that

$$\underset{j\to \mathrm{\infty }}{lim}I(u_j)=\underset{j\to \mathrm{\infty }}{lim}(I_{{\lambda }_j}(u_j)+({\lambda }_j-1){\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}H(t,u_j)dt).$$

We distinguish two cases: either $lim{sup}_{j\to \mathrm{\infty }}{I}_{{\lambda }_j}(u_j)>0$ or $lim{sup}_{j\to \mathrm{\infty }}{I}_{{\lambda }_j}(u_j)\le 0$. If the first case holds, we get $lim{sup}_{j\to \mathrm{\infty }}I(u_j)>0$ and the result follows from Lemma 2.5. If $lim{sup}_{j\to \mathrm{\infty }}{I}_{{\lambda }_j}(u_j)\le 0$, we define the sequence $\{z_j\}\in E$ by $z_j=t_j{u}_j$ with $t_j\in [0,1]$ satisfying

$$I_{{\lambda }_j}(z_j)=\underset{t\in [0,1]}{max}{I}_{{\lambda }_j}(t{u}_j).$$

(3.1)

(If for a $j\in \mathbb{N}$, $t_j$ defined by (3.1) is not unique we choose the smaller possible value.) Since $\{u_j\}$ is bounded, $\{z_j\}$ is bounded. Note that $I_{{\lambda }_j}^{\prime }(z_j)z_j=0$, $\mathrm{\forall }j\in \mathbb{N}$, thus

$${\lambda }_j{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}G(t,z_j)dt=I_{{\lambda }_j}(z_j)-\frac{1}{2}{I}_{{\lambda }_j}^{\prime }(z_j)z_j=I_{{\lambda }_j}(z_j).$$

(3.2)

On the other hand it is easily seen, following the proof of Lemma 2.1, that $I_{{\lambda }_j}^{\prime }(u)u={\parallel u\parallel }^2+o({\parallel u\parallel }^2)$ as $u\to 0$, uniformly in $j\in \mathbb{N}$. Therefore, since $I_{{\lambda }_j}^{\prime }(u_j)=0$, there is $\theta >0$ such that $\parallel u_j\parallel \ge \theta$, $\mathrm{\forall }j\in \mathbb{N}$. Recording that $lim{sup}_{j\to \mathrm{\infty }}{I}_{{\lambda }_j}(u_j)\le 0$, then we obtain from Lemma 2.1 and (3.1) $lim{inf}_{j\to \mathrm{\infty }}{I}_{{\lambda }_j}(z_j)>0$, and from (3.2) it follows that

$$lim\underset{j\to \mathrm{\infty }}{inf}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}G(t,z_j)dt=lim\underset{j\to \mathrm{\infty }}{inf}{I}_{{\lambda }_j}(z_j)>0.$$

It follows from the fact $\{z_j\}$ is bounded and Lemma 2.4 that $\{z_j\}$ does not vanish, so $\{u_j\}$ does not vanish. The proof of $u\ne 0$ and $I^{\prime }(u)=0$ is similar to the proof of Lemma 2.5. □