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

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

(VS)

where $u=(u_1,u_2,\dots ,u_N)\in {\mathbb{R}}^N$, B is an antisymmetric $N\times N$ constant matrix, $L\in C(\mathbb{R},{\mathbb{R}}^{N\times N})$ is a symmetric matrix valued function and $W\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$. As usual we say that a solution u of (VS) is homoclinic (to 0) if $u\in C^2(\mathbb{R},{\mathbb{R}}^N)$, $u\not\equiv 0$, $u(t)\to 0$, and $\dot{u}(t)\to 0$ as $|t|\to \mathrm{\infty }$.

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

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

(HS)

Inspired by the excellent monographs and works [1–3], by now, the existence and multiplicity of periodic and homoclinic solutions for HSs have extensively been investigated in many articles via variational methods, see [4–22]. Also second-order HSs with impulses via variational methods have recently been considered in [23–26]. More precisely, in 1990, Rabinowitz [3] 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 [13] introduced a type of coercive condition on the matrix L:

($L_1$) $l(t):={inf}_{|x|=1}L(t)x\cdot x\to +\mathrm{\infty }$, 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

$$0<\mu W(t,u)\le W_u(t,u(t))u,\mathrm{\forall }t\in \mathbb{R}\text{;and;}u\in {\mathbb{R}}^N\setminus \{0\},$$

where $\mu >2$ is a constant. Later, Ding [7] strengthened condition ($L_1$) by

($L_2$) there exists a constant $\alpha >0$ such that

$$l(t){|t|}^{-\alpha }\to +\mathrm{\infty }\text{as;}|t|\to \mathrm{\infty }.$$

Under the condition ($L_2$) and some subquadratic conditions on $W(t,u)$, 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 [27–29]. Zhang and Yuan [28] studied the existence of homoclinic orbits for the nonperiodic system (VS) when W satisfies the subquadratic condition at infinity. Soon after, Wu and Zhang [27] obtained the existence and multiplicity of homoclinic orbits for the nonperiodic system (VS) when W satisfies the local (AR) growth condition

$$0<\mu W(t,u)\le W_u(t,u)u,\mathrm{\forall }t\in \mathbb{R}\text{;and;}|u|\ge r,$$

(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

$$meas\{t\in \mathbb{R}:{|t|}^{-\beta }L(t)<b{I}_N\}<+\mathrm{\infty },\mathrm{\forall }b>0,$$

and

($L_4$) there exists a constant $\gamma \ge 0$ such that

$$l(t):=\underset{|x|=1}{inf}L(t)x\cdot x\ge -\gamma ,\mathrm{\forall }t\in \mathbb{R},$$

which are first used in [20]. 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 $L(t)$ satisfying ($L_3$) and ($L_4$), but not satisfying ($L_1$) or ($L_2$). For example,

$$L(t)=(t^4{sin}^{2}t+1)I_N.$$

We consider the following conditions:

($W_1$) $W\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$, and there exist positive constants $c_1$ and $\nu >2$ such that

$$c_1{|u|}^{\nu }\le W_u(t,u)u,\mathrm{\forall }(t,u)\in \mathbb{R}\times {\mathbb{R}}^N.$$

($W_2$) $W_u(t,u)=o(|u|)$ as $|u|\to 0$ uniformly in t.

($W_3$) $\tilde{W}(t,u):=\frac{1}{2}{W}_u(t,u)u-W(t,u)>0$ if $u\ne 0$, and

$$inf\{\frac{\tilde{W}(t,u)}{{|u|}^2}:t\in \mathbb{R}\text{;with;}a\le |u|<b\}>0,$$

for any $a,b>0$.

($W_4$) There exist $r>0$ and $\sigma >1$ such that ${|W_u(t,u)|}^{\sigma }\le c\tilde{W}(t,u){|u|}^{\sigma }$ 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 [27] we present the following examples. These functions satisfy the weak superquadratic conditions ($W_1$)-($W_4$), but not verify the growth condition (1).

Example:

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

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

In fact it is easy to verify that ($W_1$)-($W_4$) are satisfied. However, similar to the discussion of Remark 1.2 in [30], let $u_n={(ϵ(n\pi +\frac{3\pi }{4}))}^{\frac{1}{ϵ}}{e}_1$, where $e_1=(1,0,\dots ,0)$. Then for any $\mu >2$, one has

$$\begin{array}{rcl}{W}_u(t,u_n)u_n-\mu W(t,u_n)& =& a(t)[(p-\mu ){|u_n|}^p\\ +(p-2)(p-ϵ-\mu ){|u_n|}^{p-ϵ}{sin}^2({|u_n|}^{ϵ}/ϵ)\\ +(p-2){|u_n|}^{p}sin2({|u_n|}^{ϵ}/ϵ)]\\ =& a(t){|u_n|}^p[2-\mu +\frac{(p-2)(p-ϵ-\mu ){sin}^2({|u_n|}^{ϵ}/ϵ)}{{|u_n|}^{ϵ}}]\\ \to & -\mathrm{\infty }\text{as;}n\to \mathrm{\infty }.\end{array}$$

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 ${(C)}_c$-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 $H^1(\mathbb{R},{\mathbb{R}}^N)$ which is a Hilbert space with the inner product and norm given by

$${〈u,v〉}_H={\int }_{\mathbb{R}}[(\dot{u}(t),\dot{v}(t))+(u(t),v(t))]dt$$

and

$${\parallel u\parallel }_H={\left({\int }_{\mathbb{R}}[{|\dot{u}(t)|}^2+{|u(t)|}^2]dt\right)}^{\frac{1}{2}}$$

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

$$〈Ju,v〉={\int }_{\mathbb{R}}(Bu,\dot{v})dt$$

(2)

for all $u,v\in H$. Since B is an antisymmetric $N\times N$ constant matrix, J is self-adjoint on H. Moreover, we denote by A the self-adjoint extension of the operator $-\frac{d^2}{d{t}^2}+L(t)+J$ with the domain $\mathcal{D}(A)\subset L^2(\mathbb{R},{\mathbb{R}}^N)$. Let ${|\cdot |}_p$ be the usual $L^p$-norm, and ${〈\cdot ,\cdot 〉}_2$ the usual $L^2$-inner product. Set $E:=\mathcal{D}({|A|}^{\frac{1}{2}})$, 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 $H^1(\mathbb{R},{\mathbb{R}}^N)$. Using a similar proof of Lemma 3.1 in [20], we can prove the following lemma.

Lemma 4 Suppose that$L(t)$satisfies ($L_3$) and ($L_4$), then E is compactly embedded into$L^p(\mathbb{R},{\mathbb{R}}^N)$for$p\in [1,+\mathrm{\infty }]$.

By Lemma 4, it is easy to prove that the spectrum $\sigma (A)$ 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 ${\{e_k\}}_{k\in \mathbb{N}}$, $A{e}_k={\lambda }_k{e}_k$, form an orthogonal basis in $L^2(\mathbb{R},{\mathbb{R}}^N)$. Assume ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{{\ell }^{-}}<0$, ${\lambda }_{{\ell }^{-}+1}=\cdots ={\lambda }_{\ell }=0$ and let $E^{-}:=span\{e_1,\dots ,e_{{\ell }^{-}}\}$, $E^0:=span\{e_{{\ell }^{-}+1},\dots ,e_{\ell }\}$, and $E^{+}:={cl}_E(span\{e_{\ell +1},\dots \})$. 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 [7]). So by Lemma 4, we see that there exists a constant ${\eta }_p>0$ such that

$${|u|}_p\le {\eta }_p\parallel u\parallel ,\mathrm{\forall }u\in E,\mathrm{\forall }p\in [1,+\mathrm{\infty }].$$

Define the functional Φ on E by

$$\mathrm{\Phi }(u)={\int }_{\mathbb{R}}[\frac{1}{2}{|\dot{u}(t)|}^2+\frac{1}{2}(Bu(t),\dot{u}(t))+\frac{1}{2}(L(t)u(t),u(t))-W(t,u(t))]dt.$$

Then

$$\mathrm{\Phi }(u)=\frac{1}{2}({\parallel u^{+}\parallel }^2-{\parallel u^{-}\parallel }^2)-{\int }_{\mathbb{R}}W(t,u(t))dt,$$

(3)

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

$$\mathrm{\Psi }(u):={\int }_{\mathbb{R}}W(t,u)dt.$$

From the assumptions it follows that Φ is defined on the Banach space E and belongs to $C^1(E,\mathbb{R})$. 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 $u(t)\to 0$ and $\dot{u}(t)\to 0$, as $|t|\to \mathrm{\infty }$ (see Lemma 3.1 in [31]).

In order to study the critical points of Φ, we now recall a critical point theorem, see [32].

Let E be a Banach space. A sequence $\{u_n\}\subset E$ is said to be a ${(C)}_c$-sequence if

$$\mathrm{\Phi }(u_n)\to c\text{and}(1+\parallel u_n\parallel ){\mathrm{\Phi }}^{\prime }(u_n)\to 0.$$

Φ is said to satisfy the ${(C)}_c$-condition if any ${(C)}_c$-sequence has a convergent subsequence.

Theorem 5 ([32])

Suppose$\mathrm{\Phi }\in C^1(E,\mathbb{R})$, $E=X\oplus Y$, where$dimX<\mathrm{\infty }$, there exist$R>\rho >0$, $\kappa >0$and$e_0\in Y\setminus \{0\}$such that$inf\mathrm{\Phi }(Y\cap S_{\rho })\ge \kappa$and$sup\mathrm{\Phi }(\partial Q)\le 0$, where$S_{\rho }:=S_{\rho }(0)$is the sphere of radius ρ and center 0, and

$$Q=\{u=x+s{e}_0:s\ge 0,x\in X,\parallel u\parallel \le R\}.$$

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

Linking structure

First we discuss the linking structure of Φ. By condition ($W_1$), one has

$$W(t,u)\ge c_1{|u|}^{\nu }\ge 0,$$

(4)

for all $(t,u)\in \mathbb{R}\times {\mathbb{R}}^N$. Observe that if ($W_4$) holds, and together with (4), then if $|u|>r$, one has

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

and hence

$$|W_u(t,u)|\le {\left(\frac{c}{2}\right)}^{\frac{1}{\sigma -1}}{|u|}^{\frac{\sigma +1}{\sigma -1}},\text{if;}|u|\ge r.$$

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

$$|W_u(t,u)|\le {\left(\frac{c}{2}\right)}^{\frac{1}{\sigma -1}}{|u|}^{p-1},\text{if;}|u|\ge r.$$

(5)

Remark that $(W_2)$ and (5) imply that, for any $\epsilon >0$, there is $C_{\epsilon }>0$ such that

$$|W_u(t,u)|\le \epsilon |u|+C_{\epsilon }{|u|}^{p-1},$$

(6)

and

$$|W(t,u)|\le \epsilon {|u|}^2+C_{\epsilon }{|u|}^p,$$

(7)

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

Lemma 6 Let ($W_1$)-($W_2$) be satisfied, and assume further that$(W_4)$holds. Then there exists$\rho >0$such that$\kappa :=inf\mathrm{\Phi }(S_{\rho }^{+})>0$, where$S_{\rho }^{+}=\partial B_{\rho }\cap E^{+}$.

Proof By (7) we have

$$\mathrm{\Psi }(u)\le \epsilon {|u|}_2^2+C_{\epsilon }{|u|}_p^p\le c(\epsilon {\parallel u\parallel }^2+C_{\epsilon }{\parallel u\parallel }^p)$$

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

Denote

$$\mathcal{H}:=\mathbb{R}{e}_{\ell +1},E_{\mathcal{H}}=E^{-}\oplus E^0\oplus \mathcal{H}.$$

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$\mathrm{\Phi }(u)\le 0$for all$u\in E_{\mathcal{H}}$with$\parallel u\parallel \ge R_{E_{\mathcal{H}}}$.

Proof It suffices to show that $\mathrm{\Phi }(u)\to -\mathrm{\infty }$ 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

$${\left|u_1^{+}\right|}_2^2={〈u_1^{+},u〉}_2\le {\left|u_1^{+}\right|}_{{\nu }^{\prime }}{|u|}_{\nu }\le c{\left|u_1^{+}\right|}_2{|u|}_{\nu },$$

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

$$\begin{array}{rcl}\mathrm{\Phi }(u)& =& \frac{1}{2}{\parallel u^{+}\parallel }^2-\frac{1}{2}{\parallel u^{-}\parallel }^2-{\int }_{\mathbb{R}}W(t,u(t))dt\\ \le & c{\left|u_1^{+}\right|}_{\nu }^2-\frac{1}{2}{\parallel u^{-}\parallel }^2-c{|u_1^{+}+u^{-}+u^0|}_{\nu }^{\nu }\\ \le & c{|u_1^{+}+u^{-}+u^0|}_{\nu }^2-\frac{1}{2}{\parallel u^{-}\parallel }^2-c{|u_1^{+}+u^{-}+u^0|}_{\nu }^{\nu },\end{array}$$

which shows that $\mathrm{\Phi }(u)\to -\mathrm{\infty }$ 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$sup\mathrm{\Phi }(\partial M)\le \kappa$where$M:=\{u=u^{-}+u^0+se:u^{-}+u^0\in E^{-}\oplus E^0,s\ge 0,\parallel u\parallel \le r_1\}$and κ is given by Lemma 6.

The ${(C)}_c$-sequence

In this section, we discuss the ${(C)}_c$-sequence of Φ.

Lemma 9 Let ($L_3$)-($L_4$) and ($W_1$)-($W_4$) hold. Then any${(C)}_c$-sequence is bounded.

Proof Let $\{u_j\}\subset E$ be such that

$$\mathrm{\Phi }(u_j)\to c\text{and}(1+\parallel u_j\parallel ){\mathrm{\Phi }}^{\prime }(u_j)\to 0.$$

Then, for $C_0>0$,

$$C_0\ge \mathrm{\Phi }(u_j)-\frac{1}{2}{\mathrm{\Phi }}^{\prime }(u_j)u_j={\int }_{\mathbb{R}}\tilde{W}(t,u_j)dt.$$

(8)

Suppose to the contrary that $\{u_j\}$ 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, $y_j\rightharpoonup y$ in E, and $y_j\to y$ in $L^p$ for $p\ge 1$.

Note that

$$\begin{array}{rcl}o(1)& =& {\mathrm{\Phi }}^{\prime }(u_j)(u_j^{+}-u_j^{-})\\ =& {\parallel u_j\parallel }^2-{\int }_{\mathbb{R}}{W}_u(t,u_j)(u_j^{+}-u_j^{-})dt\\ =& {\parallel u_j\parallel }^2-{\parallel u_j\parallel }^2{\int }_{\mathbb{R}}\frac{W_u(t,u_j)(y_j^{+}-y_j^{-})}{\parallel u_j\parallel }dt\\ =& {\parallel u_j\parallel }^2(1-{\int }_{\mathbb{R}}\frac{W_u(t,u_j)(y_j^{+}-y_j^{-})}{\parallel u_j\parallel }dt).\end{array}$$

(9)

From (10), we obtain

$${\int }_{\mathbb{R}}\frac{W_u(t,u_j)(y_j^{+}-y_j^{-})}{\parallel u_j\parallel }dt\to 1.$$

(10)

Set for $s\ge 0$,

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

(11)

By ($W_1$) and ($W_3$), $h(s)>0$ for all $s>0$, and $h(s)\to \mathrm{\infty }$ as $s\to \mathrm{\infty }$.

For $0\le l<m$, let

$$C_l^m=inf\{\frac{\tilde{W}(t,u)}{{|u|}^2}:t\in \mathbb{R}\text{;with;}l\le |u(t)|<m\},$$

and

$${\mathrm{\Omega }}_j(l,m)=\{t\in \mathbb{R}:l\le |u_j(t)|<m\}.$$

(12)

Then by ($W_3$) one has $C_l^m>0$ and

$$\tilde{W}(t,u_j)\ge C_l^m{|u_j|}^2\text{for all;}t\in {\mathrm{\Omega }}_j(l,m).$$

It follows from (8) and (12) that

$$\begin{array}{rcl}{C}_0& \ge & {\int }_{{\mathrm{\Omega }}_j(0,l)}\tilde{W}(t,u_j)dt+{\int }_{{\mathrm{\Omega }}_j(l,m)}\tilde{W}(t,u_j)dt+{\int }_{{\mathrm{\Omega }}_j(m,\mathrm{\infty })}\tilde{W}(t,u_j)dt\\ \ge & {\int }_{{\mathrm{\Omega }}_j(0,l)}\tilde{W}(t,u_j)dt+C_l^m{\int }_{{\mathrm{\Omega }}_j(l,m)}{|u_j|}^{2}dt+h(m)|{\mathrm{\Omega }}_j(m,\mathrm{\infty })|.\end{array}$$

(13)

Using (13) we obtain

$$|{\mathrm{\Omega }}_j(m,\mathrm{\infty })|\le \frac{C_0}{h(m)}\to 0,$$

(14)

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

$${\int }_{{\mathrm{\Omega }}_j(l,m)}{|y_j|}^{2}dt=\frac{1}{{\parallel u_j\parallel }^2}{\int }_{{\mathrm{\Omega }}_j(l,m)}{|u_j|}^{2}dt\le \frac{C_0}{C_l^m{\parallel u_j\parallel }^2}\to 0,$$

(15)

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

$${\int }_{{\mathrm{\Omega }}_j(m,\mathrm{\infty })}{|y_j|}^{s}dt\le {\left({\int }_{{\mathrm{\Omega }}_j(m,\mathrm{\infty })}{|y_j|}^{2s}dt\right)}^{1/2}\cdot {|{\mathrm{\Omega }}_j(m,\mathrm{\infty })|}^{1/2}\le c{|{\mathrm{\Omega }}_j(m,\mathrm{\infty })|}^{1/2}\to 0,$$

(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

$$|W_u(t,u)|<\frac{ϵ}{c}|u|$$

for all $|u|\le l_{ϵ}$. Consequently,

$$\begin{array}{rcl}{\int }_{{\mathrm{\Omega }}_j(0,l_{ϵ})}\frac{W_u(t,u_j)(y_j^{+}-y_j^{-})|y_j|}{|u_j|}dt& \le & {\int }_{{\mathrm{\Omega }}_j(0,l_{ϵ})}\frac{ϵ}{c}|y_j^{+}-y_j^{-}||y_j|dt\\ \le & \frac{ϵ}{c}{|y_j|}_2^2<ϵ\end{array}$$

(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 $C=C(ϵ)>0$ independent of j such that $|W_u(t,u_j)|\le C|u_j|$ for $t\in {\mathrm{\Omega }}_j(l_{ϵ},m_{ϵ})$. By (15) there is $j_0$ such that

$$\begin{array}{rcl}{\int }_{{\mathrm{\Omega }}_j(l_{ϵ},m_{ϵ})}\frac{W_u(t,u_j)(y_j^{+}-y_j^{-})|y_j|}{|u_j|}dt& \le & C{\int }_{{\mathrm{\Omega }}_j(l_{ϵ},m_{ϵ})}|y_j^{+}-y_j^{-}||y_j|dt\\ \le & C{|y_j|}_2{\left({\int }_{{\mathrm{\Omega }}_j(l_{ϵ},m_{ϵ})}{|y_j|}^{2}dt\right)}^{1/2}\\ \le & ϵ\end{array}$$

(19)

for all $j\ge j_0$. By (17)-(19), one has

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_{\mathbb{R}}\frac{W_u(t,u_j)(y_j^{+}-y_j^{-})}{\parallel u_j\parallel }dt\le 3ϵ<1,$$

(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 [33]. Clearly, by assumptions, $\mathrm{\Psi }(u)\ge 0$. Let $u_j\rightharpoonup u$ in E. By Lemma 10, $u_j\to u$ in $L^p(\mathbb{R})$ for $p\ge 2$, and $u_j(t)\to u(t)$ a.e. $t\in \mathbb{R}$. Hence $W(t,u_j)\to W(t,u)$ for a.e. $t\in \mathbb{R}$. Thus, it follows from Fatou’s lemma that

$$\mathrm{\Psi }(u)={\int }_{\mathbb{R}}W(t,u)dt={\int }_{\mathbb{R}}\underset{j\to \mathrm{\infty }}{lim}W(t,u_j)dt\le \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\int }_{\mathbb{R}}W(t,u_j)dt=\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\mathrm{\Psi }(u_j),$$

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 $\phi \in C_0^{\mathrm{\infty }}(\mathbb{R})$,

$${\mathrm{\Psi }}^{\prime }(u_j)\phi ={\int }_{\mathbb{R}}{W}_u(t,u_j)\phi dt\to {\int }_{\mathbb{R}}{W}_u(t,u)\phi dt={\mathrm{\Psi }}^{\prime }(u)\phi .$$

(21)

Since $C_0^{\mathrm{\infty }}(\mathbb{R})$ is dense in E, for any $v\in E$, we take ${\phi }_n\in C_0^{\mathrm{\infty }}(\mathbb{R})$ such that

$$\parallel {\phi }_n-v\parallel \to 0\text{as;}j\to \mathrm{\infty }.$$

By (6), one has

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

For any $ϵ>0$, fix n so that $\parallel v-{\phi }_n\parallel <ϵ/2c$. By (21) there exists $j_0$ such that

$$\left|({\mathrm{\Psi }}^{\prime }(u_j)-{\mathrm{\Psi }}^{\prime }(u)){\phi }_n\right|<ϵ/2\text{for all;}j\ge j_0.$$

Then $|({\mathrm{\Psi }}^{\prime }(u_j)-{\mathrm{\Psi }}^{\prime }(u)){\phi }_n|<ϵ$ 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 $u_j\rightharpoonup u$ in E, then ${\mathrm{\Phi }}^{\prime }(u_j)\to {\mathrm{\Phi }}^{\prime }(u)$. Let $\{u_j\}$ be an arbitrary ${(C)}_c$-sequence, by Lemma 9, it is bounded, up to a subsequence, we may assume $u_j\rightharpoonup u$ in E. Plainly, u is a critical point of Φ.

Lemma 11 Under the assumptions of Lemma 9, Φ satisfies${(C)}_c$-condition.

Proof Let $\{u_j\}$ be any ${(C)}_c$-sequence. By Lemmas 4, 9, and 10, one has

and

$$\begin{array}{rcl}o(1)& =& ({\mathrm{\Phi }}^{\prime }(u_j)-{\mathrm{\Phi }}^{\prime }(u),u_j^{+}-u^{+})\\ =& {\parallel u_j^{+}-u^{+}\parallel }^2+{\int }_{\mathbb{R}}(W_u(t,u_j)-W_u(t,u))(u_j^{+}-u^{+})dt\\ =& {\parallel u_j^{+}-u^{+}\parallel }^2+o(1).\end{array}$$

So $u_j^{+}\to u^{+}$ as $j\to \mathrm{\infty }$. Since $dim(E^{-}\oplus E^0)<\mathrm{\infty }$, 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 ${(C)}_c$-condition. Therefore, by Theorem 5 Φ has at least one critical point u. □