Theory and Modern Applications

# On homoclinic orbits for a class of damped vibration systems

## Abstract

In this article, we establish the new result on homoclinic orbits for a class of damped vibration systems. Some recent results in the literature are generalized and significantly improved.

MSC:49J40, 70H05.

## Introduction and main results

Consider the following second-order damped vibration problems

$\stackrel{¨}{u}\left(t\right)+B\stackrel{˙}{u}\left(t\right)-L\left(t\right)u\left(t\right)+{W}_{u}\left(t,u\left(t\right)\right)=0,\phantom{\rule{1em}{0ex}}t\in \mathbb{R},$
(VS)

where $u=\left({u}_{1},{u}_{2},\dots ,{u}_{N}\right)\in {\mathbb{R}}^{N}$, B is an antisymmetric $N×N$ constant matrix, $L\in C\left(\mathbb{R},{\mathbb{R}}^{N×N}\right)$ is a symmetric matrix valued function and $W\in {C}^{1}\left(\mathbb{R}×{\mathbb{R}}^{N},\mathbb{R}\right)$. As usual we say that a solution u of (VS) is homoclinic (to 0) if $u\in {C}^{2}\left(\mathbb{R},{\mathbb{R}}^{N}\right)$, $u\not\equiv 0$, $u\left(t\right)\to 0$, and $\stackrel{˙}{u}\left(t\right)\to 0$ as $|t|\to \mathrm{\infty }$.

When B is a zero matrix, (VS) is just the following second-order Hamiltonian systems (HSs)

$\stackrel{¨}{u}\left(t\right)-L\left(t\right)u\left(t\right)+{W}_{u}\left(t,u\left(t\right)\right)=0,\phantom{\rule{1em}{0ex}}t\in \mathbb{R}.$
(HS)

Inspired by the excellent monographs and works , by now, the existence and multiplicity of periodic and homoclinic solutions for HSs have extensively been investigated in many articles via variational methods, see . Also second-order HSs with impulses via variational methods have recently been considered in . More precisely, in 1990, Rabinowitz  established the existence result on homoclinic orbit for the periodic second-order HS. It is well known that the periodicity is used to control the lack of compactness due to the fact that HS is set on all $\mathbb{R}$.

For the nonperiodic case, the problem is quite different from the one described in nature. Rabinowitz and Tanaka  introduced a type of coercive condition on the matrix L:

(${L}_{1}$) $l\left(t\right):={inf}_{|x|=1}L\left(t\right)x\cdot x\to +\mathrm{\infty }$, as $|t|\to \mathrm{\infty }$.

They established a compactness lemma under the nonperiodic case and obtained the existence of homoclinic orbit for the nonperiodic system (HS) under the usual Ambrosetti-Rabinowitz (AR) growth condition

$0<\mu W\left(t,u\right)\le {W}_{u}\left(t,u\left(t\right)\right)u,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \mathbb{R}\text{;and;}u\in {\mathbb{R}}^{N}\setminus \left\{0\right\},$

where $\mu >2$ is a constant. Later, Ding  strengthened condition (${L}_{1}$) by

(${L}_{2}$) there exists a constant $\alpha >0$ such that

$l\left(t\right){|t|}^{-\alpha }\to +\mathrm{\infty }\phantom{\rule{1em}{0ex}}\text{as;}|t|\to \mathrm{\infty }.$

Under the condition (${L}_{2}$) and some subquadratic conditions on $W\left(t,u\right)$, Ding proved the existence and multiplicity of homoclinic orbits for the system (HS). From then on, the condition (${L}_{1}$) or (${L}_{2}$) are extensively used in many articles.

Compared with the case where B is a zero matrix, the case where $B\ne 0$, i.e., the nonperiodic system (VS), has been considered only by a few authors, see . Zhang and Yuan  studied the existence of homoclinic orbits for the nonperiodic system (VS) when W satisfies the subquadratic condition at infinity. Soon after, Wu and Zhang  obtained the existence and multiplicity of homoclinic orbits for the nonperiodic system (VS) when W satisfies the local (AR) growth condition

$0<\mu W\left(t,u\right)\le {W}_{u}\left(t,u\right)u,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \mathbb{R}\text{;and;}|u|\ge r,$
(1)

where $\mu >2$ and $r>0$ are two constants. It is worth noticing that the matrix L is required to satisfy the condition (${L}_{1}$) in the above two articles.

Inspired by [27, 28], in this article we shall replace the condition (${L}_{1}$) on L by the following conditions:

(${L}_{3}$) there exists a constant $\beta >1$ such that

$meas\left\{t\in \mathbb{R}:{|t|}^{-\beta }L\left(t\right)0,$

and

(${L}_{4}$) there exists a constant $\gamma \ge 0$ such that

$l\left(t\right):=\underset{|x|=1}{inf}L\left(t\right)x\cdot x\ge -\gamma ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \mathbb{R},$

which are first used in . By using a recent critical point theorem, we prove that the nonperiodic system (VS) has at least one homoclinic orbit when W satisfies weak superquadratic at the infinity, which improve and extend the results of [27, 28].

Remark 1 In fact, there are some matrix-valued functions $L\left(t\right)$ satisfying (${L}_{3}$) and (${L}_{4}$), but not satisfying (${L}_{1}$) or (${L}_{2}$). For example,

$L\left(t\right)=\left({t}^{4}{sin}^{2}t+1\right){I}_{N}.$

We consider the following conditions:

(${W}_{1}$) $W\in {C}^{1}\left(\mathbb{R}×{\mathbb{R}}^{N},\mathbb{R}\right)$, and there exist positive constants ${c}_{1}$ and $\nu >2$ such that

${c}_{1}{|u|}^{\nu }\le {W}_{u}\left(t,u\right)u,\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(t,u\right)\in \mathbb{R}×{\mathbb{R}}^{N}.$

(${W}_{2}$) ${W}_{u}\left(t,u\right)=o\left(|u|\right)$ as $|u|\to 0$ uniformly in t.

(${W}_{3}$) $\stackrel{˜}{W}\left(t,u\right):=\frac{1}{2}{W}_{u}\left(t,u\right)u-W\left(t,u\right)>0$ if $u\ne 0$, and

$inf\left\{\frac{\stackrel{˜}{W}\left(t,u\right)}{{|u|}^{2}}:t\in \mathbb{R}\text{;with;}a\le |u|0,$

for any $a,b>0$.

(${W}_{4}$) There exist $r>0$ and $\sigma >1$ such that ${|{W}_{u}\left(t,u\right)|}^{\sigma }\le c\stackrel{˜}{W}\left(t,u\right){|u|}^{\sigma }$ if $|u|\ge r$.

Theorem 2 Assume that (${L}_{3}$)-(${L}_{4}$) and (${W}_{1}$)-(${W}_{4}$) hold. Then the system (VS) has at least one homoclinic orbit.

Remark 3 To see that our result generalizes  we present the following examples. These functions satisfy the weak superquadratic conditions (${W}_{1}$)-(${W}_{4}$), but not verify the growth condition (1).

Example:

$W\left(t,u\right)=a\left(t\right)\left({|u|}^{p}+\left(p-2\right){|u|}^{p-ϵ}{sin}^{2}\left(\frac{{|u|}^{ϵ}}{ϵ}\right)\right),$

where ${inf}_{t\in \mathbb{R}}a\left(t\right)>0$, and $p>2$, $0<ϵ.

In fact it is easy to verify that (${W}_{1}$)-(${W}_{4}$) are satisfied. However, similar to the discussion of Remark 1.2 in , let ${u}_{n}={\left(ϵ\left(n\pi +\frac{3\pi }{4}\right)\right)}^{\frac{1}{ϵ}}{e}_{1}$, where ${e}_{1}=\left(1,0,\dots ,0\right)$. Then for any $\mu >2$, one has

$\begin{array}{rcl}{W}_{u}\left(t,{u}_{n}\right){u}_{n}-\mu W\left(t,{u}_{n}\right)& =& a\left(t\right)\left[\left(p-\mu \right){|{u}_{n}|}^{p}\\ +\left(p-2\right)\left(p-ϵ-\mu \right){|{u}_{n}|}^{p-ϵ}{sin}^{2}\left({|{u}_{n}|}^{ϵ}/ϵ\right)\\ +\left(p-2\right){|{u}_{n}|}^{p}sin2\left({|{u}_{n}|}^{ϵ}/ϵ\right)\right]\\ =& a\left(t\right){|{u}_{n}|}^{p}\left[2-\mu +\frac{\left(p-2\right)\left(p-ϵ-\mu \right){sin}^{2}\left({|{u}_{n}|}^{ϵ}/ϵ\right)}{{|{u}_{n}|}^{ϵ}}\right]\\ \to & -\mathrm{\infty }\phantom{\rule{1em}{0ex}}\text{as;}n\to \mathrm{\infty }.\end{array}$

That is, the condition (1) is not satisfied for any $\mu >2$.

This article is organized as follows. In the following section, we formulate the variational setting and recall a critical point theorem required. In section ‘Linking structure’, we discuss linking structure of the functional. In section ‘The ${\left(C\right)}_{c}$-sequence’, we study the Cerami condition of the functional and give the proof of Theorem 2.

Notation Throughout the article, we shall denote by $c>0$ various positive constants which may vary from line to line and are not essential to the problem.

## Variational setting

In this section, we establish a variational setting for the system (VS). Let H be ${H}^{1}\left(\mathbb{R},{\mathbb{R}}^{N}\right)$ which is a Hilbert space with the inner product and norm given by

${〈u,v〉}_{H}={\int }_{\mathbb{R}}\left[\left(\stackrel{˙}{u}\left(t\right),\stackrel{˙}{v}\left(t\right)\right)+\left(u\left(t\right),v\left(t\right)\right)\right]\phantom{\rule{0.2em}{0ex}}dt$

and

${\parallel u\parallel }_{H}={\left({\int }_{\mathbb{R}}\left[{|\stackrel{˙}{u}\left(t\right)|}^{2}+{|u\left(t\right)|}^{2}\right]\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{2}}$

for $u,v\in H$, where $\left(\cdot ,\cdot \right)$ denotes the inner product in ${\mathbb{R}}^{N}$. It is well known that H is continuously embedded in ${L}^{p}\left(\mathbb{R},{\mathbb{R}}^{N}\right)$ for $p\in \left[2,\mathrm{\infty }\right)$. Define an operator $J:H\to H$ by

$〈Ju,v〉={\int }_{\mathbb{R}}\left(Bu,\stackrel{˙}{v}\right)\phantom{\rule{0.2em}{0ex}}dt$
(2)

for all $u,v\in H$. Since B is an antisymmetric $N×N$ constant matrix, J is self-adjoint on H. Moreover, we denote by A the self-adjoint extension of the operator $-\frac{{d}^{2}}{d{t}^{2}}+L\left(t\right)+J$ with the domain $\mathcal{D}\left(A\right)\subset {L}^{2}\left(\mathbb{R},{\mathbb{R}}^{N}\right)$. Let ${|\cdot |}_{p}$ be the usual ${L}^{p}$-norm, and ${〈\cdot ,\cdot 〉}_{2}$ the usual ${L}^{2}$-inner product. Set $E:=\mathcal{D}\left({|A|}^{\frac{1}{2}}\right)$, the domain of ${|A|}^{\frac{1}{2}}$. Define on E the inner product

${〈u,v〉}_{E}:={〈{|A|}^{\frac{1}{2}}u,{|A|}^{\frac{1}{2}}v〉}_{2}+{〈u,v〉}_{2}$

and the norm

${\parallel u\parallel }_{E}={〈u,u〉}_{E}^{\frac{1}{2}}.$

Then E is a Hilbert space and it is easy to verify that E is continuously embedded in ${H}^{1}\left(\mathbb{R},{\mathbb{R}}^{N}\right)$. Using a similar proof of Lemma 3.1 in , we can prove the following lemma.

Lemma 4 Suppose that$L\left(t\right)$satisfies (${L}_{3}$) and (${L}_{4}$), then E is compactly embedded into${L}^{p}\left(\mathbb{R},{\mathbb{R}}^{N}\right)$for$p\in \left[1,+\mathrm{\infty }\right]$.

By Lemma 4, it is easy to prove that the spectrum $\sigma \left(A\right)$ has a sequence of eigenvalues (counted with their multiplicities)

${\lambda }_{1}\le {\lambda }_{2}\le \cdots \le {\lambda }_{k}\le \cdots$

with ${\lambda }_{k}\to +\mathrm{\infty }$ as $k\to +\mathrm{\infty }$, and corresponding eigenfunctions ${\left\{{e}_{k}\right\}}_{k\in \mathbb{N}}$, $A{e}_{k}={\lambda }_{k}{e}_{k}$, form an orthogonal basis in ${L}^{2}\left(\mathbb{R},{\mathbb{R}}^{N}\right)$. Assume ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{{\ell }^{-}}<0$, ${\lambda }_{{\ell }^{-}+1}=\cdots ={\lambda }_{\ell }=0$ and let ${E}^{-}:=span\left\{{e}_{1},\dots ,{e}_{{\ell }^{-}}\right\}$, ${E}^{0}:=span\left\{{e}_{{\ell }^{-}+1},\dots ,{e}_{\ell }\right\}$, and ${E}^{+}:={cl}_{E}\left(span\left\{{e}_{\ell +1},\dots \right\}\right)$. Then

$E={E}^{-}\oplus {E}^{0}\oplus {E}^{+}$

is an orthogonal decomposition of E. We introduce on E the following product

$〈u,v〉:={〈{|A|}^{\frac{1}{2}}u,{|A|}^{\frac{1}{2}}v〉}_{2}+{〈{u}^{0},{v}^{0}〉}_{2},$

and the norm

$\parallel u\parallel ={〈u,u〉}^{\frac{1}{2}},$

where $u={u}^{-}+{u}^{0}+{u}^{+}$, $v={v}^{-}+{v}^{0}+{v}^{+}\in {E}^{-}\oplus {E}^{0}\oplus {E}^{+}$. Then $\parallel \cdot \parallel$ and ${\parallel \cdot \parallel }_{E}$ are equivalent (see ). So by Lemma 4, we see that there exists a constant ${\eta }_{p}>0$ such that

${|u|}_{p}\le {\eta }_{p}\parallel u\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u\in E,\mathrm{\forall }p\in \left[1,+\mathrm{\infty }\right].$

Define the functional Φ on E by

$\mathrm{\Phi }\left(u\right)={\int }_{\mathbb{R}}\left[\frac{1}{2}{|\stackrel{˙}{u}\left(t\right)|}^{2}+\frac{1}{2}\left(Bu\left(t\right),\stackrel{˙}{u}\left(t\right)\right)+\frac{1}{2}\left(L\left(t\right)u\left(t\right),u\left(t\right)\right)-W\left(t,u\left(t\right)\right)\right]\phantom{\rule{0.2em}{0ex}}dt.$

Then

$\mathrm{\Phi }\left(u\right)=\frac{1}{2}\left({\parallel {u}^{+}\parallel }^{2}-{\parallel {u}^{-}\parallel }^{2}\right)-{\int }_{\mathbb{R}}W\left(t,u\left(t\right)\right)\phantom{\rule{0.2em}{0ex}}dt,$
(3)

where $u={u}^{-}+{u}^{0}+{u}^{+}\in E$. Furthermore, define

$\mathrm{\Psi }\left(u\right):={\int }_{\mathbb{R}}W\left(t,u\right)\phantom{\rule{0.2em}{0ex}}dt.$

From the assumptions it follows that Φ is defined on the Banach space E and belongs to ${C}^{1}\left(E,\mathbb{R}\right)$. A standard argument shows that critical points of Φ are solutions of the system (VS). Moreover, it is easy to verify that if $u\not\equiv 0$ is a solution of (VS), then $u\left(t\right)\to 0$ and $\stackrel{˙}{u}\left(t\right)\to 0$, as $|t|\to \mathrm{\infty }$ (see Lemma 3.1 in ).

In order to study the critical points of Φ, we now recall a critical point theorem, see .

Let E be a Banach space. A sequence $\left\{{u}_{n}\right\}\subset E$ is said to be a ${\left(C\right)}_{c}$-sequence if

$\mathrm{\Phi }\left({u}_{n}\right)\to c\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left(1+\parallel {u}_{n}\parallel \right){\mathrm{\Phi }}^{\prime }\left({u}_{n}\right)\to 0.$

Φ is said to satisfy the ${\left(C\right)}_{c}$-condition if any ${\left(C\right)}_{c}$-sequence has a convergent subsequence.

Theorem 5 ()

Suppose$\mathrm{\Phi }\in {C}^{1}\left(E,\mathbb{R}\right)$, $E=X\oplus Y$, where$dimX<\mathrm{\infty }$, there exist$R>\rho >0$, $\kappa >0$and${e}_{0}\in Y\setminus \left\{0\right\}$such that$inf\mathrm{\Phi }\left(Y\cap {S}_{\rho }\right)\ge \kappa$and$sup\mathrm{\Phi }\left(\partial Q\right)\le 0$, where${S}_{\rho }:={S}_{\rho }\left(0\right)$is the sphere of radius ρ and center 0, and

$Q=\left\{u=x+s{e}_{0}:s\ge 0,x\in X,\parallel u\parallel \le R\right\}.$

Moreover, if Φ satisfies the${\left(C\right)}_{c}$-condition for all$c\in \left[\kappa ,sup\mathrm{\Phi }\left(Q\right)\right]$, then Φ has a critical value in$\left[\kappa ,sup\mathrm{\Phi }\left(Q\right)\right]$.

First we discuss the linking structure of Φ. By condition (${W}_{1}$), one has

$W\left(t,u\right)\ge {c}_{1}{|u|}^{\nu }\ge 0,$
(4)

for all $\left(t,u\right)\in \mathbb{R}×{\mathbb{R}}^{N}$. Observe that if (${W}_{4}$) holds, and together with (4), then if $|u|>r$, one has

$\begin{array}{rcl}{|{W}_{u}\left(t,u\right)|}^{\sigma }& \le & c\left(\frac{1}{2}{W}_{u}\left(t,u\right)u-W\left(t,u\right)\right){|u|}^{\sigma }\\ \le & \frac{c}{2}{W}_{u}\left(t,u\right)u{|u|}^{\sigma }\\ \le & \frac{c}{2}|{W}_{u}\left(t,u\right)|{|u|}^{\sigma +1},\end{array}$

and hence

$|{W}_{u}\left(t,u\right)|\le {\left(\frac{c}{2}\right)}^{\frac{1}{\sigma -1}}{|u|}^{\frac{\sigma +1}{\sigma -1}},\phantom{\rule{1em}{0ex}}\text{if;}|u|\ge r.$

Let $p=2\sigma /\left(\sigma -1\right)>2$. Then we have

$|{W}_{u}\left(t,u\right)|\le {\left(\frac{c}{2}\right)}^{\frac{1}{\sigma -1}}{|u|}^{p-1},\phantom{\rule{1em}{0ex}}\text{if;}|u|\ge r.$
(5)

Remark that $\left({W}_{2}\right)$ and (5) imply that, for any $\epsilon >0$, there is ${C}_{\epsilon }>0$ such that

$|{W}_{u}\left(t,u\right)|\le \epsilon |u|+{C}_{\epsilon }{|u|}^{p-1},$
(6)

and

$|W\left(t,u\right)|\le \epsilon {|u|}^{2}+{C}_{\epsilon }{|u|}^{p},$
(7)

for all $\left(t,u\right)\in \mathbb{R}×{\mathbb{R}}^{N}$.

Lemma 6 Let (${W}_{1}$)-(${W}_{2}$) be satisfied, and assume further that$\left({W}_{4}\right)$holds. Then there exists$\rho >0$such that$\kappa :=inf\mathrm{\Phi }\left({S}_{\rho }^{+}\right)>0$, where${S}_{\rho }^{+}=\partial {B}_{\rho }\cap {E}^{+}$.

Proof By (7) we have

$\mathrm{\Psi }\left(u\right)\le \epsilon {|u|}_{2}^{2}+{C}_{\epsilon }{|u|}_{p}^{p}\le c\left(\epsilon {\parallel u\parallel }^{2}+{C}_{\epsilon }{\parallel u\parallel }^{p}\right)$

for all $u\in E$, the lemma follows from the form of Φ (see (3)). □

Denote

$\mathcal{H}:=\mathbb{R}{e}_{\ell +1},\phantom{\rule{2em}{0ex}}{E}_{\mathcal{H}}={E}^{-}\oplus {E}^{0}\oplus \mathcal{H}.$

Then ${E}_{\mathcal{H}}$ is a finite subspace.

Lemma 7 Under the assumptions of Theorem 2, there exists${R}_{{E}_{\mathcal{H}}}>0$such that$\mathrm{\Phi }\left(u\right)\le 0$for all$u\in {E}_{\mathcal{H}}$with$\parallel u\parallel \ge {R}_{{E}_{\mathcal{H}}}$.

Proof It suffices to show that $\mathrm{\Phi }\left(u\right)\to -\mathrm{\infty }$ in ${E}_{\mathcal{H}}$ as $\parallel u\parallel \to \mathrm{\infty }$. For any $u\in {E}_{\mathcal{H}}$, let $u={u}_{1}^{+}+{u}^{-}+{u}^{0}$, where ${u}_{1}^{+}\in \mathcal{H}$, ${u}^{-}\in {E}^{-}$, ${u}^{0}\in {E}^{0}$. Since $dim\mathcal{H}=1$, then

${|{u}_{1}^{+}|}_{2}^{2}={〈{u}_{1}^{+},u〉}_{2}\le {|{u}_{1}^{+}|}_{{\nu }^{\prime }}{|u|}_{\nu }\le c{|{u}_{1}^{+}|}_{2}{|u|}_{\nu },$

where $\frac{1}{{\nu }^{\prime }}+\frac{1}{\nu }=1$. Thus ${|{u}_{1}^{+}|}_{\nu }\le c{|{u}_{1}^{+}|}_{2}\le c{|u|}_{\nu }$, and together with (4), we obtain

$\begin{array}{rcl}\mathrm{\Phi }\left(u\right)& =& \frac{1}{2}{\parallel {u}^{+}\parallel }^{2}-\frac{1}{2}{\parallel {u}^{-}\parallel }^{2}-{\int }_{\mathbb{R}}W\left(t,u\left(t\right)\right)\phantom{\rule{0.2em}{0ex}}dt\\ \le & c{|{u}_{1}^{+}|}_{\nu }^{2}-\frac{1}{2}{\parallel {u}^{-}\parallel }^{2}-c{|{u}_{1}^{+}+{u}^{-}+{u}^{0}|}_{\nu }^{\nu }\\ \le & c{|{u}_{1}^{+}+{u}^{-}+{u}^{0}|}_{\nu }^{2}-\frac{1}{2}{\parallel {u}^{-}\parallel }^{2}-c{|{u}_{1}^{+}+{u}^{-}+{u}^{0}|}_{\nu }^{\nu },\end{array}$

which shows that $\mathrm{\Phi }\left(u\right)\to -\mathrm{\infty }$ as $\parallel u\parallel \to \mathrm{\infty }$. □

As a special case we have

Lemma 8 Assume that the assumptions of Theorem 2 are satisfied. Then, letting$e\in \mathcal{H}$with$\parallel e\parallel =1$, there is${r}_{1}>\rho >0$such that$sup\mathrm{\Phi }\left(\partial M\right)\le \kappa$where$M:=\left\{u={u}^{-}+{u}^{0}+se:{u}^{-}+{u}^{0}\in {E}^{-}\oplus {E}^{0},s\ge 0,\parallel u\parallel \le {r}_{1}\right\}$and κ is given by Lemma 6.

## The ${\left(C\right)}_{c}$-sequence

In this section, we discuss the ${\left(C\right)}_{c}$-sequence of Φ.

Lemma 9 Let (${L}_{3}$)-(${L}_{4}$) and (${W}_{1}$)-(${W}_{4}$) hold. Then any${\left(C\right)}_{c}$-sequence is bounded.

Proof Let $\left\{{u}_{j}\right\}\subset E$ be such that

$\mathrm{\Phi }\left({u}_{j}\right)\to c\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left(1+\parallel {u}_{j}\parallel \right){\mathrm{\Phi }}^{\prime }\left({u}_{j}\right)\to 0.$

Then, for ${C}_{0}>0$,

${C}_{0}\ge \mathrm{\Phi }\left({u}_{j}\right)-\frac{1}{2}{\mathrm{\Phi }}^{\prime }\left({u}_{j}\right){u}_{j}={\int }_{\mathbb{R}}\stackrel{˜}{W}\left(t,{u}_{j}\right)\phantom{\rule{0.2em}{0ex}}dt.$
(8)

Suppose to the contrary that $\left\{{u}_{j}\right\}$ is unbounded. Setting ${y}_{j}={u}_{j}/\parallel {u}_{j}\parallel$, then $\parallel {y}_{j}\parallel =1$, ${|{y}_{j}|}_{p}\le c\parallel {y}_{j}\parallel =c$ for all $p\ge 2$. Passing to subsequence, ${y}_{j}⇀y$ in E, and ${y}_{j}\to y$ in ${L}^{p}$ for $p\ge 1$.

Note that

$\begin{array}{rcl}o\left(1\right)& =& {\mathrm{\Phi }}^{\prime }\left({u}_{j}\right)\left({u}_{j}^{+}-{u}_{j}^{-}\right)\\ =& {\parallel {u}_{j}\parallel }^{2}-{\int }_{\mathbb{R}}{W}_{u}\left(t,{u}_{j}\right)\left({u}_{j}^{+}-{u}_{j}^{-}\right)\phantom{\rule{0.2em}{0ex}}dt\\ =& {\parallel {u}_{j}\parallel }^{2}-{\parallel {u}_{j}\parallel }^{2}{\int }_{\mathbb{R}}\frac{{W}_{u}\left(t,{u}_{j}\right)\left({y}_{j}^{+}-{y}_{j}^{-}\right)}{\parallel {u}_{j}\parallel }\phantom{\rule{0.2em}{0ex}}dt\\ =& {\parallel {u}_{j}\parallel }^{2}\left(1-{\int }_{\mathbb{R}}\frac{{W}_{u}\left(t,{u}_{j}\right)\left({y}_{j}^{+}-{y}_{j}^{-}\right)}{\parallel {u}_{j}\parallel }\phantom{\rule{0.2em}{0ex}}dt\right).\end{array}$
(9)

From (10), we obtain

${\int }_{\mathbb{R}}\frac{{W}_{u}\left(t,{u}_{j}\right)\left({y}_{j}^{+}-{y}_{j}^{-}\right)}{\parallel {u}_{j}\parallel }\phantom{\rule{0.2em}{0ex}}dt\to 1.$
(10)

Set for $s\ge 0$,

$h\left(s\right):=inf\left\{\stackrel{˜}{W}\left(t,u\right):t\in \mathbb{R}\text{;and;}u\in {\mathbb{R}}^{N}\text{;with;}|u|\ge s\right\}.$
(11)

By (${W}_{1}$) and (${W}_{3}$), $h\left(s\right)>0$ for all $s>0$, and $h\left(s\right)\to \mathrm{\infty }$ as $s\to \mathrm{\infty }$.

For $0\le l, let

${C}_{l}^{m}=inf\left\{\frac{\stackrel{˜}{W}\left(t,u\right)}{{|u|}^{2}}:t\in \mathbb{R}\text{;with;}l\le |u\left(t\right)|

and

${\mathrm{\Omega }}_{j}\left(l,m\right)=\left\{t\in \mathbb{R}:l\le |{u}_{j}\left(t\right)|
(12)

Then by (${W}_{3}$) one has ${C}_{l}^{m}>0$ and

$\stackrel{˜}{W}\left(t,{u}_{j}\right)\ge {C}_{l}^{m}{|{u}_{j}|}^{2}\phantom{\rule{1em}{0ex}}\text{for all;}t\in {\mathrm{\Omega }}_{j}\left(l,m\right).$

It follows from (8) and (12) that

$\begin{array}{rcl}{C}_{0}& \ge & {\int }_{{\mathrm{\Omega }}_{j}\left(0,l\right)}\stackrel{˜}{W}\left(t,{u}_{j}\right)\phantom{\rule{0.2em}{0ex}}dt+{\int }_{{\mathrm{\Omega }}_{j}\left(l,m\right)}\stackrel{˜}{W}\left(t,{u}_{j}\right)\phantom{\rule{0.2em}{0ex}}dt+{\int }_{{\mathrm{\Omega }}_{j}\left(m,\mathrm{\infty }\right)}\stackrel{˜}{W}\left(t,{u}_{j}\right)\phantom{\rule{0.2em}{0ex}}dt\\ \ge & {\int }_{{\mathrm{\Omega }}_{j}\left(0,l\right)}\stackrel{˜}{W}\left(t,{u}_{j}\right)\phantom{\rule{0.2em}{0ex}}dt+{C}_{l}^{m}{\int }_{{\mathrm{\Omega }}_{j}\left(l,m\right)}{|{u}_{j}|}^{2}\phantom{\rule{0.2em}{0ex}}dt+h\left(m\right)|{\mathrm{\Omega }}_{j}\left(m,\mathrm{\infty }\right)|.\end{array}$
(13)

Using (13) we obtain

$|{\mathrm{\Omega }}_{j}\left(m,\mathrm{\infty }\right)|\le \frac{{C}_{0}}{h\left(m\right)}\to 0,$
(14)

as $m\to \mathrm{\infty }$ uniformly in j, and for any fixed $0,

${\int }_{{\mathrm{\Omega }}_{j}\left(l,m\right)}{|{y}_{j}|}^{2}\phantom{\rule{0.2em}{0ex}}dt=\frac{1}{{\parallel {u}_{j}\parallel }^{2}}{\int }_{{\mathrm{\Omega }}_{j}\left(l,m\right)}{|{u}_{j}|}^{2}\phantom{\rule{0.2em}{0ex}}dt\le \frac{{C}_{0}}{{C}_{l}^{m}{\parallel {u}_{j}\parallel }^{2}}\to 0,$
(15)

as $j\to \mathrm{\infty }$. It follows from (14) that, for any $s\in \left[2,+\mathrm{\infty }\right)$,

${\int }_{{\mathrm{\Omega }}_{j}\left(m,\mathrm{\infty }\right)}{|{y}_{j}|}^{s}\phantom{\rule{0.2em}{0ex}}dt\le {\left({\int }_{{\mathrm{\Omega }}_{j}\left(m,\mathrm{\infty }\right)}{|{y}_{j}|}^{2s}\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/2}\cdot {|{\mathrm{\Omega }}_{j}\left(m,\mathrm{\infty }\right)|}^{1/2}\le c{|{\mathrm{\Omega }}_{j}\left(m,\mathrm{\infty }\right)|}^{1/2}\to 0,$
(16)

as $m\to \mathrm{\infty }$ uniformly in j.

Let $0<ϵ<\frac{1}{3}$. By (${W}_{2}$) there is ${l}_{ϵ}>0$ such that

$|{W}_{u}\left(t,u\right)|<\frac{ϵ}{c}|u|$

for all $|u|\le {l}_{ϵ}$. Consequently,

$\begin{array}{rcl}{\int }_{{\mathrm{\Omega }}_{j}\left(0,{l}_{ϵ}\right)}\frac{{W}_{u}\left(t,{u}_{j}\right)\left({y}_{j}^{+}-{y}_{j}^{-}\right)|{y}_{j}|}{|{u}_{j}|}\phantom{\rule{0.2em}{0ex}}dt& \le & {\int }_{{\mathrm{\Omega }}_{j}\left(0,{l}_{ϵ}\right)}\frac{ϵ}{c}|{y}_{j}^{+}-{y}_{j}^{-}||{y}_{j}|\phantom{\rule{0.2em}{0ex}}dt\\ \le & \frac{ϵ}{c}{|{y}_{j}|}_{2}^{2}<ϵ\end{array}$
(17)

for all j.

Set ${\sigma }^{\prime }:=p/2$. By (${W}_{4}$), (16) and Hölder inequality, we can take ${m}_{ϵ}\ge r$ large enough such that (18)

for all j. Note that there is $C=C\left(ϵ\right)>0$ independent of j such that $|{W}_{u}\left(t,{u}_{j}\right)|\le C|{u}_{j}|$ for $t\in {\mathrm{\Omega }}_{j}\left({l}_{ϵ},{m}_{ϵ}\right)$. By (15) there is ${j}_{0}$ such that

$\begin{array}{rcl}{\int }_{{\mathrm{\Omega }}_{j}\left({l}_{ϵ},{m}_{ϵ}\right)}\frac{{W}_{u}\left(t,{u}_{j}\right)\left({y}_{j}^{+}-{y}_{j}^{-}\right)|{y}_{j}|}{|{u}_{j}|}\phantom{\rule{0.2em}{0ex}}dt& \le & C{\int }_{{\mathrm{\Omega }}_{j}\left({l}_{ϵ},{m}_{ϵ}\right)}|{y}_{j}^{+}-{y}_{j}^{-}||{y}_{j}|\phantom{\rule{0.2em}{0ex}}dt\\ \le & C{|{y}_{j}|}_{2}{\left({\int }_{{\mathrm{\Omega }}_{j}\left({l}_{ϵ},{m}_{ϵ}\right)}{|{y}_{j}|}^{2}\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/2}\\ \le & ϵ\end{array}$
(19)

for all $j\ge {j}_{0}$. By (17)-(19), one has

$\underset{j\to \mathrm{\infty }}{lim sup}{\int }_{\mathbb{R}}\frac{{W}_{u}\left(t,{u}_{j}\right)\left({y}_{j}^{+}-{y}_{j}^{-}\right)}{\parallel {u}_{j}\parallel }\phantom{\rule{0.2em}{0ex}}dt\le 3ϵ<1,$
(20)

which contradicts with (10). The proof is complete. □

Lemma 10 Under the assumptions of Theorem 2, Ψ is nonnegative, weakly sequentially lower semi-continuous, and${\mathrm{\Psi }}^{\prime }$is weakly sequentially continuous. Moreover, ${\mathrm{\Psi }}^{\prime }$is compact.

Proof We follow the idea of . Clearly, by assumptions, $\mathrm{\Psi }\left(u\right)\ge 0$. Let ${u}_{j}⇀u$ in E. By Lemma 10, ${u}_{j}\to u$ in ${L}^{p}\left(\mathbb{R}\right)$ for $p\ge 2$, and ${u}_{j}\left(t\right)\to u\left(t\right)$ a.e. $t\in \mathbb{R}$. Hence $W\left(t,{u}_{j}\right)\to W\left(t,u\right)$ for a.e. $t\in \mathbb{R}$. Thus, it follows from Fatou’s lemma that

$\mathrm{\Psi }\left(u\right)={\int }_{\mathbb{R}}W\left(t,u\right)\phantom{\rule{0.2em}{0ex}}dt={\int }_{\mathbb{R}}\underset{j\to \mathrm{\infty }}{lim}W\left(t,{u}_{j}\right)\phantom{\rule{0.2em}{0ex}}dt\le \underset{j\to \mathrm{\infty }}{lim inf}{\int }_{\mathbb{R}}W\left(t,{u}_{j}\right)\phantom{\rule{0.2em}{0ex}}dt=\underset{j\to \mathrm{\infty }}{lim inf}\mathrm{\Psi }\left({u}_{j}\right),$

which shows that the function Ψ is weakly sequentially lower semi-continuous.

Now we show that ${\mathrm{\Psi }}^{\prime }$ is compact. It is clear that, for any $\phi \in {C}_{0}^{\mathrm{\infty }}\left(\mathbb{R}\right)$,

${\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)\phi ={\int }_{\mathbb{R}}{W}_{u}\left(t,{u}_{j}\right)\phi \phantom{\rule{0.2em}{0ex}}dt\to {\int }_{\mathbb{R}}{W}_{u}\left(t,u\right)\phi \phantom{\rule{0.2em}{0ex}}dt={\mathrm{\Psi }}^{\prime }\left(u\right)\phi .$
(21)

Since ${C}_{0}^{\mathrm{\infty }}\left(\mathbb{R}\right)$ is dense in E, for any $v\in E$, we take ${\phi }_{n}\in {C}_{0}^{\mathrm{\infty }}\left(\mathbb{R}\right)$ such that

$\parallel {\phi }_{n}-v\parallel \to 0\phantom{\rule{1em}{0ex}}\text{as;}j\to \mathrm{\infty }.$

By (6), one has

$\begin{array}{rcl}|{\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)v-{\mathrm{\Psi }}^{\prime }\left(u\right)v|& \le & |\left({\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\right){\phi }_{n}|+|\left({\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\right)\left(v-{\phi }_{n}\right)|\\ \le & |\left({\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\right){\phi }_{n}|\\ +c{\int }_{\mathbb{R}}\left(|u|+|{u}_{j}|+{|u|}^{p-1}+{|{u}_{j}|}^{p-1}\right)|v-{\phi }_{n}|\\ \le & |\left({\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\right){\phi }_{n}|+c\parallel v-{\phi }_{n}\parallel .\end{array}$

For any $ϵ>0$, fix n so that $\parallel v-{\phi }_{n}\parallel <ϵ/2c$. By (21) there exists ${j}_{0}$ such that

$|\left({\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\right){\phi }_{n}|<ϵ/2\phantom{\rule{1em}{0ex}}\text{for all;}j\ge {j}_{0}.$

Then $|\left({\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\right){\phi }_{n}|<ϵ$ for all $j\ge {j}_{0}$, which proves the weakly sequentially continuity. Therefore, ${\mathrm{\Psi }}^{\prime }$ is compact by the weakly continuity of ${\mathrm{\Psi }}^{\prime }$ since E is a Hilbert space. □

Lemma 10 implies that ${\mathrm{\Phi }}^{\prime }$ is weakly sequentially continuous, i.e., if ${u}_{j}⇀u$ in E, then ${\mathrm{\Phi }}^{\prime }\left({u}_{j}\right)\to {\mathrm{\Phi }}^{\prime }\left(u\right)$. Let $\left\{{u}_{j}\right\}$ be an arbitrary ${\left(C\right)}_{c}$-sequence, by Lemma 9, it is bounded, up to a subsequence, we may assume ${u}_{j}⇀u$ in E. Plainly, u is a critical point of Φ.

Lemma 11 Under the assumptions of Lemma 9, Φ satisfies${\left(C\right)}_{c}$-condition.

Proof Let $\left\{{u}_{j}\right\}$ be any ${\left(C\right)}_{c}$-sequence. By Lemmas 4, 9, and 10, one has and

$\begin{array}{rcl}o\left(1\right)& =& \left({\mathrm{\Phi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Phi }}^{\prime }\left(u\right),{u}_{j}^{+}-{u}^{+}\right)\\ =& {\parallel {u}_{j}^{+}-{u}^{+}\parallel }^{2}+{\int }_{\mathbb{R}}\left({W}_{u}\left(t,{u}_{j}\right)-{W}_{u}\left(t,u\right)\right)\left({u}_{j}^{+}-{u}^{+}\right)\phantom{\rule{0.2em}{0ex}}dt\\ =& {\parallel {u}_{j}^{+}-{u}^{+}\parallel }^{2}+o\left(1\right).\end{array}$

So ${u}_{j}^{+}\to {u}^{+}$ as $j\to \mathrm{\infty }$. Since $dim\left({E}^{-}\oplus {E}^{0}\right)<\mathrm{\infty }$, we have ${u}_{j}^{-}+{u}_{j}^{0}\to {u}^{-}+{u}^{0}$, and therefore ${u}_{j}\to u$ as $j\to \mathrm{\infty }$ in E. □

## Proof of the theorem

Proof of Theorem 2 Lemma 8 shows that Φ possesses the linking structure of Theorem 5, and Lemma 11 implies that Φ satisfies the ${\left(C\right)}_{c}$-condition. Therefore, by Theorem 5 Φ has at least one critical point u. □

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## Acknowledgement

The research of J. J. Nieto was partially supported by the Ministerio de Economía y Competitividad and FEDER, project MTM2010-15314. The research of J. Sun was supported by the National Natural Science Foundation of China (Grant No. 11201270, 11271372), Shandong Natural Science Foundation (Grant No. ZR2012AQ010).

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Correspondence to Juntao Sun.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Each of the authors, JS, JN and MO contributed to each part of this study equally and read and approved the final version of the manuscript.

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Sun, J., Nieto, J.J. & Otero-Novoa, M. On homoclinic orbits for a class of damped vibration systems. Adv Differ Equ 2012, 102 (2012). https://doi.org/10.1186/1687-1847-2012-102

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### Keywords

• homoclinic orbits
• second-order systems
• damped vibration problems
• variational methods
• ${\left(C\right)}_{c}$-sequence 