New homoclinic orbits for Hamiltonian systems with asymptotically quadratic growth at infinity

TL;DR

This paper establishes the existence and multiplicity of homoclinic solutions in Hamiltonian systems with asymptotically quadratic nonlinearities, introducing new conditions and embedding theorems to extend prior results.

Contribution

It introduces a new coercive condition and embedding theorem to prove the existence of multiple homoclinic orbits in such Hamiltonian systems.

Findings

At least one nontrivial homoclinic orbit exists.

Infinitely many homoclinic orbits are found under symmetry conditions.

New asymptotically quadratic conditions differ from previous studies.

Abstract

In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity \begin{eqnarray*} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0. {eqnarray*} We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by Generalized Mountain Pass Theorem. By Variant Fountain Theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.

Equations

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

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

$$l(t)=\underset{|x|=1}{\inf }(L(t)x,x)\to +\infty \text{mbox}as|t|\to \infty .$$

$$\underset{|s|\to +\infty }{\lim }\text{mbox}meas\left(\left\{t\in (s-r_0,s+r_0):L(t)\ngeqq y{I}_N\right\}\right)=0,\forall y>0.$$

$$\underset{t\in \mathbb{R},c_1\le |x|\le c_2}{\inf }\frac{\tilde{W}(t,x)}{|x{|}^2}<0$$

$$\nabla W(t,x)=a(t)x+\nabla G(t,x),$$

$$\underset{t\in \mathbb{R}}{\inf }\left[\underset{|x|=1}{\inf }(a(t)x,x)\right]>\inf (\sigma (\mathcal{H})\cap (0,+\infty )),$$

$$\underset{R\to +\infty }{\lim }\text{mbox}meas\left(\left\{|t|\ge R:\frac{L(t)}{|t|{\ln }^{\alpha }|t|}\le M{I}_N\right\}\right)=0.$$

$$(u,w)=(|\mathcal{H}{|}^{1/2}u,|\mathcal{H}{|}^{1/2}w{)}_2+(u^0,w^0{)}_2\text{mbox}and\Vert u\Vert =(u,u{)}^{1/2},$$

$$|\nabla W(t,x)|\le b_1|x|\forall (t,x)\in \mathbb{R}\times {\mathbb{R}}^N;$$

$$\tilde{W}(t,x)\ge b_2|x{|}^{\mu }\text{mbox}if|x|\ge r_1\forall t\in \mathbb{R};$$

$$\tilde{W}(t,x)\ge -b_3|x|\forall t\in \mathbb{R}\text{mbox}and|x|<r_1;$$

$$|W(t,x)|\le \frac{{\lambda }_{\bar{n}+1}-{\sigma }_0}{2}|x{|}^2\text{mbox}forany|x|\le r_2\text{mbox}andt\in \mathbb{R};$$

$$W(t,x)\ge \left(\frac{{\lambda }_{\bar{n}+1}}{2}+{\epsilon }_0\right)|x{|}^2\text{mbox}forany|x|\ge r_{\infty }\text{mbox}andt\in \mathbb{R};$$

$$I(u)$$

$$\Vert u{\Vert }_{L^p}\le C_p\Vert u\Vert ,\forall u\in E.$$

$${\Gamma }_R=\left\{t\in \mathbb{R}:|t|\le R\right\}$$

$$D_M(R)=\left\{t\in {\Gamma }_R^c:\frac{L(t)}{|t|{\ln }^{\alpha }|t|}\le M{I}_N\right\}.$$

$$\Vert u{\Vert }_o^2={\int }_{\mathbb{R}}\left(|\dot{u}(t){|}^2+(L(t)u(t),u(t))+|u(t){|}^2\right)dt.$$

$${\int }_{{\Gamma }_R}|v_i(t)|dt\le \frac{1}{2}\epsilon ,$$

$${\int }_{{\Gamma }_R^c}|v_i(t)|dt\le {\int }_{D_M(R)}|v_i(t)|dt+{\int }_{{\Gamma }_R^c\setminus D_M(R)}|v_i(t)|dt.$$

$${\int }_{{\Gamma }_R^c\setminus D_M(R)}|v_i(t)|dt$$

$${\int }_{{\Gamma }_R^c}|v_i(t)|dt$$

$$u(t)={\int }_t^{t+1}(-\dot{u}(\tau )(t+1-\tau {)}^{n+1}+u(\tau )(n+1)(t+1-\tau {)}^n)d\tau .$$

$$|u(t)|\le \frac{1}{\sqrt{2n+3}}{\left({\int }_t^{t+1}|\dot{u}(\tau ){|}^{2}d\tau \right)}^{\frac{1}{2}}+(n+1)\left({\int }_t^{t+1}|u(\tau )|d\tau \right).$$

$$|u(t)-v(t)|$$

$$\underset{t\in {\Gamma }_{R+1}^c}{\max }|u(t)-v(t)|\le \epsilon .$$

$$\Vert u{\Vert }_{\sim }^2={\Vert (\mathcal{A}+a{)}^{\frac{1}{2}}u\Vert }_2^2+\Vert u{\Vert }_2^2.$$

$$\displaystyle\langle I^{\prime}(u),u\rangle=\|u^{+}\|^{2}-\|u^{-}\|^{2}-\int_{\mathbb{R}}(\nabla W(t,u(t)),u(t)))dt,\ \ \mbox{ $\forall u$ $\in E$}.$$

$$|I(u_n)|\le M_3,\Vert I^{\prime }(u_n)\Vert \le M_3.$$

Full text

New homoclinic orbits for Hamiltonian systems with asymptotically quadratic growth at infinity††thanks: D.-L. Wu is supported by NSF of China (No.11801472), China Scholarship Council (No.201708515186)

and the Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No.2017CXTD02). X. Yu is supported by NSF of China (No.11701464) and the Fundamental Research Funds for the Central Universities (No.JBK1805001).

Dong-Lun Wua,b, Xiang Yuc111Corresponding author.

aCollege of Science, Southwest Petroleum University,

Chengdu, Sichuan 610500, P.R. China

bInstitute of Nonlinear Dynamics, Southwest Petroleum University,

Chengdu, Sichuan 610500, P.R. China

cSchool of Economic and Mathematics, Southwestern University of Finance and Economics,

Chengdu, Sichuan 611130, P.R. China

††footnotetext: Email address: wudl2008@163.com; yuxiang@swufe.edu.cn

Abstract In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity

[TABLE]

We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by Generalized Mountain Pass Theorem. By Variant Fountain Theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.

Keywords Homoclinic solutions; Asymptotically quadratic Hamiltonian systems; Embedding theorem; Generalized Mountain Pass Theorem; Variant Fountain Theorem.

1 Introduction

In this paper, we consider the following systems

[TABLE]

where $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$ and $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric matrix. A solution $u(t)$ of (1) is called nontrivial homoclinic if $u\not\equiv 0$, $u(t)\rightarrow 0$ as $t\rightarrow\pm\infty$. Moreover, given a $N\times N$ matrix $\mathcal{M}$, we say $\mathcal{M}\geq 0$ if $\inf_{|x|=1}(\mathcal{M}x,x)\geq 0$ and $\mathcal{M}\ngeq 0$ if $\mathcal{M}\geq 0$ does not hold.

In last decades, along with the development of variational methods, many mathematicians showed the existence and multiplicity of solutions for differential equations(see[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]). One of the difficulties to obtain homoclinic orbits for (1) is the lack of compactness of embeddings. To solve this problem, Rabinowitz and his co-authors introduced the periodic and coercive conditions. With the periodic assumption, Rabinowitz [13] obtained homoclinic solutions for (1) by taking limit of a sequence of subharmonic orbits. After then there are many papers concerning on the existence and multiplicity of homoclinic solutions for problem (1) under periodic assumption. Without periodic assumptions, Rabinowitz and Tanaka [14] introduced following coercive condition on $L(t)$ to retrieve the compactness.

$(L^{\prime})$ Let $L$ be a positively definite matrix and

[TABLE]

Omana and Willem [9] used condition $(L^{\prime})$ to obtain a compact embedding theorem. In 2010, Wang, Zhang and Xu [25] introduced the following new coercive condition.

$(L^{\prime\prime})$ The measure of the set $\Lambda^{b}:=\{t\in\mathbb{R}:L(t)<bI_{N}\}$ is finite for some $b>0$, where $I_{N}$ is the identity matrix of order $N$.

Under condition $(L^{\prime\prime})$, the spectrum of the operator $-(d^{2}/dt^{2})+L(t)$ can go to $-\infty$ and the corresponding functional of problem (1) becomes strong indefinite, i.e. unbounded from below and from above on infinite dimensional spaces. Another coercive condition can be found in [7] as follow.

$(L^{\prime\prime\prime})$ Form some $r_{0}>0$,

[TABLE]

In 1995, in order to get the homoclinic solutions for subquadratic Hamiltonian systems, Ding [4] introduced the following condition.

$(L^{\prime\prime\prime\prime})$ From some $\varsigma<1$, $l(t)|t|^{\varsigma-2}\rightarrow+\infty$ as $|t|\rightarrow\infty$.

As far as we know, in [4], Ding has considered the case $L(t)$ is not positively definite for the first time. Combining with the coercive condition, $L(t)$ possesses finite negative eigenvalues. Using similar conditions, in [3], Chen obtained the ground state homoclinic solution for (1) when 0 lies in the gap of spectrum of $-(d^{2}/dt^{2})+L(t)$. Recently, Schechter [15] studied problem (1) with the conditions weak enough to make the linear operator have the essential spectrum possibly and the author considered different cases according to the number of the negative eigenvalues.

As we know, the growth of nonlinear term $W$ is important in showing the geometric structure of the corresponding functional and the boundedness of the asymptotic critical points sequence. The growth of $W$ is mainly classified into superquadratic, subquadratic and asymptotically quadratic cases. In this paper, we consider problem (1) with a set of new asymptotically quadratic growth conditions at infinity. There are already many results concerning on homoclinic solutions for the Hamiltonian systems with asymptotically quadratic growth. By investigating the references, we can find that, when dealing with the asymptotically quadratic Hamiltonian systems, some mathematicians considered the following condition.

$(G1)$ $\widetilde{W}(t,x)<0$ if $x\neq 0$ and $\widetilde{W}(t,x)\rightarrow-\infty$ as $|x|\rightarrow\infty$ uniformly in $t$, and

[TABLE]

for all $0<c_{1}<c_{2}<+\infty$, where $\widetilde{W}(t,x)=2W(t,x)-(\nabla W(t,x),x)$.

Let $\mathcal{H}$ be the self-adjoint extension of $-(d^{2}/dt^{2})+L(t)$ with $\mathfrak{D}(\mathcal{H})\subset L^{2}(\mathbb{R},\mathbb{R}^{N})$. Some other mathematicians [8] assumed that

$(G2)$ Let $W$ satisfy

[TABLE]

where $a(t)$ satisfies

[TABLE]

where $\sigma(\mathcal{H})$ the spectrum of $\mathcal{H}$ and $G(t,x)$ is subquadratic at infinity in $x$.

By above conditions and some other auxiliary conditions, mathematicians obtained the homoclinic solutions for (1). More details can be found in [2, 4, 5, 8, 15, 16, 17, 21, 24, 27, 28]. However, in these papers, the authors required that

$(G3)$ $\widetilde{W}(t,x)\leq 0$ for all $t\in\mathbb{R}$ with $|x|$ large enough.

In this paper, we consider the nonlinearities with asymptotic quadratic growth at infinity with $(G3)$ being not satisfied. To regain the compactness, we introduce the following new coercive condition.

$(L1)$ there exist $\alpha>1$ and $M>0$ such that

[TABLE]

Besides, we assume the following condition.

$(L2)$ there exists $L_{0}>0$ such that $(L(t)x,x)\geq-L_{0}|x|^{2}$ for all $(t,x)\in\mathbb{R}\times\mathbb{R}^{N}$.

By $(L1)$ and $(L2)$, similar to [4], we can set $E=\mathfrak{D}(|\mathcal{H}|^{1/2})$ be the domain of $|\mathcal{H}|^{1/2}$ and the inner product and norm on $E$ are $(u,w)_{o}=(|\mathcal{H}|^{1/2}u,|\mathcal{H}|^{1/2}w)_{2}+(u,w)_{2}$, which shows that $E$ is a Hilbert space. It can be proved that $\sigma(\mathcal{H})$ consists of a sequence of eigenvalues $\lambda_{1}\leq\lambda_{2}\leq\cdots\rightarrow\infty$, and a sequence of eigenfunctions (e_{n})$$(\mathcal{H}e_{n}=\lambda_{n}e_{n}) forms an orthogonal basis in $L^{2}$, where $\sigma(\mathcal{H})$ is the spectrum of $\mathcal{H}$. Let $n^{-}$, $n^{0}$ and $\bar{n}$ stand for the numbers of the negative, null and nonpositive eigenvalues respectively. Set $E^{-}$, $E^{0}$ and $E^{+}$ be the negative, null and positive space spanned with negative, null and positive eigenvectors respectively. Then $E=E^{-}\oplus E^{0}\oplus E^{+}$. We can define

[TABLE]

where $u=u^{-}+u^{0}+u^{+}$ and $w=w^{-}+w^{0}+w^{+}\in E$. Clearly, $\|\cdot\|_{o}$ is equivalent to $\|\cdot\|$ according to the proof of Lemma 2.1 in [4]. Our main results are as follow.

Theorem 1.1**.**

Let $(L1)$, $(L2)$ hold and $W(t,x)$ satisfy

$(W1)$* $W(t,0)=0$, $\forall t\in\mathbb{R}$, there exists $b_{1}>0$ such that*

[TABLE]

$(W2)$* there exist $b_{2}$, $r_{1}>0$ and $\mu>1$ such that*

[TABLE]

$(W3)$* there exists $b_{3}>0$ such that*

[TABLE]

$(W4)$* there exist $r_{2}$, $\sigma_{0}>0$ such that*

[TABLE]

$(W5)$* there exist $\varepsilon_{0}>0$, $r_{\infty}>0$ such that*

[TABLE]

$(W6)$* $W(t,x)\geq 0$ for all $(t,x)\in\mathbb{R}\times\mathbb{R}^{N}.$*

Then there exists a homoclinic solution for (1).

Theorem 1.2**.**

Let $(L1)$-$(L2)$, $(W1)$-$(W6)$ hold and $W(t,x)$ is even in $x$, then there are infinitely many homoclinic solutions for (1).

Remark 1**.**

Compared to condition $(L^{\prime\prime\prime\prime})$, $l(t)$ may not have limit under conditions $(L1)$ and $(L2)$. Moreover, our conditions are also different from $(L^{\prime})$, $(L^{\prime\prime})$ and $(L^{\prime\prime\prime})$.

Remark 2**.**

In our theorems, we introduce a set of new asymptotically quadratic growth conditions where the potentials $W(t,x)$ does not satisfy $(G3)$.

Remark 3**.**

As we know, there are many papers in which the infinitely many homoclinic solutions are obtained for superquadratic and subquadratic Hamiltonian systems. However, there are only few similar results for (1) under asymptotically quadratic growth conditions. In Theorem 1.2, we conclude that problem (1) has infinitely many homoclinic solutions under asymptotically quadratic growth condition without periodic conditions.

2 Preliminaries

Similar to [4], let $I:E\rightarrow\mathbb{R}$ be as follow.

[TABLE]

Lemma 2.1**.**

Assume $(L1)$ and $(L2)$ hold, then $E\hookrightarrow L^{p}(\mathbb{R},\mathbb{R}^{N})$ is compact for any $p\in[1,+\infty]$ and there is $C_{p}>0$ such that

[TABLE]

Proof. We adopt some ideas from [1] and [4]. For $R>0$, set

[TABLE]

and

[TABLE]

Since $\|\cdot\|_{o}$ is equivalent to $\|\cdot\|$, we use the norm $\|\cdot\|_{o}$ to show the proof. First, we suppose that $l(t)\geq 1$ for all $t\in\mathbb{R}$. Then $|\mathcal{H}|=\mathcal{H}$ and

[TABLE]

Let $K\subset E$ satisfy $\|u\|\leq M_{1}$ for all $u\in K$ for some $M_{1}>0$. Next, we show $K$ is precompact in $L^{p}(\mathbb{R},\mathbb{R}^{N})$ for any $p\in[1,+\infty]$.

First, we put $p=1$. By Soblolev embedding compact theorem, for any $\varepsilon>0$, there are $u_{1}$, $\ldots$, $u_{m}\in K$ such that for $\forall u\in K$, we can find $u_{i}$($i\in[1,m]$) such that

[TABLE]

where $v_{i}(t)=u(t)-u_{i}(t)$. Obviously, one has

[TABLE]

Letting $1<\eta<\alpha$ and $\Omega_{n}=\{t\in\mathbb{R}:(|t|\ln^{\eta}|t|)|v_{i}|\leq 1\}$, we obtain that

[TABLE]

From (4) and (5), there is $M_{2}>0$ such that

[TABLE]

From $(L1)$, we can choose $R$ large enough such that $\int_{\{t\in\mathbb{R}:|t|\geq R\}}|v_{i}(t)|dt<\frac{1}{2}\varepsilon$. Then we can see $K$ has a finite $\varepsilon$-net. Therefore, $K$ is precompact in $L^{1}(\mathbb{R},\mathbb{R}^{N})$.

Second, we put $p=+\infty$. For $\forall n\in\mathbb{N}$ and $t\in\mathbb{R}$, we have

[TABLE]

Then one has

[TABLE]

For any $u$, $v\in K$ with $|t|\geq R+1$, where $R\geq r_{0}$, it follows from $(L1)$, $(L2)$ and (6) that

[TABLE]

Then there exists $n_{0}$ such that $\frac{2\sqrt{2}M_{1}}{\sqrt{2n+3}}<\frac{1}{3}\varepsilon$ for any $n\geq n_{0}$. Then we can choose $R$ large enough such that the last two terms in above inequality are both smaller than $\frac{1}{3}\varepsilon$. Then one can deduce that

[TABLE]

Similar to the case $p=1$, $K$ is precompact in $L^{\infty}(\mathbb{R},\mathbb{R}^{N})$.

Finally, we will show our conclusion holds without $l(t)\geq 1$. From $(L1)$, we can see $L(t)$ is bounded from below. Hence there exists a constant $a>0$ such that $l(t)+a\geq 1$ for all $t\in\mathbb{R}$. On $\mathfrak{D}((\mathcal{A}+a)^{\frac{1}{2}})$, we introduce a norm

[TABLE]

Then it follows from the proof of Lemma 2.1 in [4] that $\|\cdot\|_{\sim}$ is equivalent to $\|\cdot\|_{o}$ and $\|\cdot\|$. We obtain our conclusion. $\Box$

Lemma 2.2**.**

Under the conditions of Theorem 1.1, we see $I$ is of $C^{1}$ class and

[TABLE]

Proof. The proof is similar to Lemma 2.2 in [28]. $\Box$

Next, we show problem (1) has at least a nontrivial homoclinic solution by the Generalized Mountain Pass Theorem under $(PS)$ condition (see Theorem 5.29 in [12]). First, we show $I$ satisfies the $(PS)$ condition.

Lemma 2.3**.**

Suppose $(L1)$-$(L2)$ and $(W1)$-$(W6)$ hold, then any $(PS)$ sequence of $I$ is bounded.

Proof. Assume that $\{u_{n}\}\subset E$ is a $(PS)$ sequence, then there is a $M_{3}>0$ such that

[TABLE]

Now we show the boundedness of $\{u_{n}\}$. We adopt an indirect argument. Assuming that $\|u_{n}\|\rightarrow+\infty$ as $n\rightarrow\infty$, then one can deduce from (7), $(W2)$ and $(W3)$ that

[TABLE]

Let

[TABLE]

and

[TABLE]

Then we have

[TABLE]

Since $E^{-}\oplus E^{0}$ is of finite dimension, there is $M_{4}>0$ such that

[TABLE]

Moreover, there exist $M_{5}$, $M_{6}>0$ such that

[TABLE]

Then one has

[TABLE]

Choosing $0<s<\min\left\{1,\frac{\mu}{2},\frac{\mu}{2(\mu-1)}\right\}$, we can deduce from (3) and (8) that

[TABLE]

for some $M_{7}$, $M_{8}$, $M_{9}>0$. By (7), we obtain

[TABLE]

By $(W1)$, (11) and (12), we deduce

[TABLE]

for some $M_{10}$, $M_{11}>0$. Hence one obtains

[TABLE]

Formula (10) and (13) show a contradiction. Therefore $\{u_{n}\}$ is bounded. $\Box$

Lemma 2.4**.**

Suppose $(L1)$-$(L2)$ and $(W1)$-$(W6)$ hold, then we can find $\alpha$, $\varrho>0$ such that $I\mid_{S}\geq\alpha$, where $S$ is a sphere in $E^{+}$ of radius $\varrho$, i.e. $S=\{u\in E^{+}|\ \|u\|=\varrho\}$.

Proof. Set

[TABLE]

One sees that $0<\|u\|_{\infty}\leq r_{2}$, $\forall u\in S$. By (2) and $(W4)$, one can see that

[TABLE]

By the definitions of $\varrho$ and $\alpha$,  (14) implies $I\mid_{S}\geq\alpha$. We obtain our conclusion.                                   $\Box$

Lemma 2.5**.**

Suppose $(L1)$-$(L2)$ and $(W1)$-$(W6)$ hold, then there exists $\rho>\varrho>0$ such that $\sup_{\partial Q}I(u)\leq 0$, where

[TABLE]

Proof. It follows from $(W5)$ and $(W6)$ that

[TABLE]

for $(t,x)\in\mathbb{R}\times\mathbb{R}^{N}$. By the definition of $Q$, for any $u\in Q$, we have

[TABLE]

for some $0\leq\zeta\leq\rho$ and $v\in E^{-}\oplus E^{0}$ with $\|v\|\leq\rho$. It can be deduced from (2), (15) and (16) that

[TABLE]

for some $M_{12}$, $M_{13}>0$ and any $u\in Q$. By the definition of $Q$, we shows

[TABLE]

where

[TABLE]

Therefore, we deduce from $(W6)$ that $\sup_{u\in Q_{1}}I(u)\leq 0$. By (17), one has

[TABLE]

which implies that $\sup_{u\in Q_{2}}I(u)\leq 0$ for $\rho$ large enough. Finally, one can deduce from (17) that

[TABLE]

for some $M_{14}>0$ since $\dim(E^{-}\oplus E^{0})<\infty$, which implies that $\sup_{u\in Q_{3}}I(u)\leq 0$ for $\rho$ large enough. Then we obtain our conclusion.                                                                                              $\Box$

Proof of Theorem 1.1. From Lemmas 2.4, 2.5 and the Generalized Mountain Pass theorem, there exists a sequence $\{u_{k}\}$ such that that $\{I(u_{k})\}$ is bounded, $I^{\prime}(u_{k})\rightarrow 0$ as $k\rightarrow\infty$. It can be obtained from Lemma 2.3 that $\{u_{k}\}$ is a bounded sequence in $E$. Then we can find a subsequence, which is still denoted by $\{u_{k}\}$, such that $u_{k}\rightharpoonup u_{0}$ in $E$, which implies that $u_{k}\rightarrow u_{0}$ in $L^{2}(\mathbb{R},\mathbb{R}^{N})$. One can imply that

[TABLE]

Then $u_{k}\rightarrow u_{0}$ as $n\rightarrow\infty$ since $E^{-}\oplus E^{0}$ is of finite dimensions. Therefore, we have $I^{\prime}(u_{0})=0$. Finally, we need to prove that $u_{0}\in\mathfrak{D}(\mathcal{H})$. It is easy to see that $u_{0}\in C^{2}$ satisfies (1). Then by (1), (3) and $(W_{1})$, we have

[TABLE]

Thus $u_{0}\in\mathfrak{D}(\mathcal{H})$. We obtain our conclusion.                                                                               $\Box$

3 Proof of Theorem 1.2

Subsequently, it will be shown that $I$ possesses infinitely many critical points if $W(t,x)$ is even in $x$, which are gotten by the Variant Fountain Theorem by Zou in [29]. For any $j\in\mathbb{N}$, define $X_{j}=span\{e_{j}\}$. Set $Y_{k}=\bigoplus_{j=1}^{k}X_{j}$, $Z_{k}=\overline{\bigoplus_{j=k}^{\infty}X_{j}}$, $B_{k}$ be a ball in $Y_{k}$ with radius $\rho_{k}$ and centered at origin. Let $N_{k}$ be the boundary of a ball in $Z_{k}$ with radius $r_{k}$ and centered at origin for $\rho_{k}>r_{k}>0$. Obviously, we have $E=\overline{\bigoplus_{j\in\mathbb{N}}X_{j}}$. Next, we show that $I$ satisfies the geometrical structure of the Variant Fountain Theorem. Set

[TABLE]

and

[TABLE]

Suppose that $I_{\lambda}(u)$ satisfies

$(A_{1})$ $I_{\lambda}$ is a bounded map uniformly in $\lambda\in[1,2]$. And $I_{\lambda}(-u)=I_{\lambda}(u)$ for $\forall(\lambda,u)\in[1,2]\times E$;

$(A_{2})$ $P(u)$ is nonnegative on $E$; $T(u)$ or $P(u)$ is coercive; or

$(A_{2}^{\prime})$ $P(u)$ is nonpositive on $E$; $-P(u)$ is coercive.

For $k>2$, let

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Lemma 3.1**.**

(Zou[29]) Assume $(A_{1})$ and $(A_{2})$(or $(A_{2}^{\prime})$). If $b_{k}(\lambda)>a_{k}(\lambda)$ for $\forall\lambda\in[1,2]$, then $c_{k}(\lambda)\geq b_{k}(\lambda)$ for $\forall\lambda\in[1,2]$. Furthermore, for a.e. $\lambda\in[1,2]$, there is $\{u_{n}^{k}(\lambda)\}_{n=1}^{\infty}$ such that

[TABLE]

Lemma 3.2**.**

Suppose $(L1)$-$(L2)$, $(W1)$-$(W6)$ hold, then we can find $r_{k}>0$, $\overline{b}_{k}\rightarrow+\infty$ as $k\rightarrow\infty$ such that $b_{k}(\lambda)\geq\overline{b}_{k}$ for $\forall\lambda\in[1,2]$.

Proof. For $k\in\mathbb{N}$ and $p\in[1,+\infty]$, we set

[TABLE]

Obviously, $\beta_{k}(p)\rightarrow 0^{+}$ as $k\rightarrow\infty$. Then, one has $Z_{k}\subset E^{+}$ and $\beta_{k}(2)\leq\frac{1}{4(\lambda_{\bar{n}+1}-\sigma_{0})}$ for $k$ large enough. By $(W1)$ and $(W4)$, we conclude that

[TABLE]

for some $M_{15}>0$ and $q>2$. For any $u\in Z_{k}$, we can deduce that

[TABLE]

Let

[TABLE]

Obviously, we have $r_{k}\rightarrow+\infty$ as $k\rightarrow\infty$. When $k$ large enough, we can conclude

[TABLE]

Then $\overline{b}_{k}\rightarrow+\infty$ as $k\rightarrow\infty$.                                                                                                   $\Box$

Lemma 3.3**.**

We can find $\rho_{k}$ big enough such that $a_{k}(\lambda)\leq 0$ for $\forall\lambda\in[1,2]$.

Proof. For $\forall u\in Y_{k}\setminus\{0\}$, $\xi\in(0,1)$ and $\delta>0$, set

[TABLE]

Subsequently, we prove that there exists $\varepsilon_{1}>0$ such that

[TABLE]

Otherwise, there exists $\{u_{n}\}_{n\in\mathbb{N}}$ such that

[TABLE]

Set $v_{n}=\frac{|u_{n}|}{\|u_{n}\|}$, then $\|v_{n}\|=1$ for $\forall n\in\mathbb{N}$, which implies that there exists $v_{0}\in Y_{k}$ such that $\|v_{0}\|=1$ and $v_{n}\rightarrow v_{0}$ in $L^{2}(\mathbb{R},\mathbb{R}^{N})$. Then one obtains

[TABLE]

Furthermore, we can obtain that there exist $\tau_{1}$, $\tau_{2}>0$ such that

[TABLE]

If not, we have $\mbox{meas}(\Gamma_{\frac{1}{n}}(v_{0}))=0$ for $\forall n\in\mathbb{N}$. Then one gets

[TABLE]

which contradicts $\|v_{0}\|=1$. Then (23) holds. For $n$ large enough, we can deduce that

[TABLE]

for all $t\in\Gamma_{\frac{1}{n}}^{c}(v_{n})\bigcap\Gamma_{\tau_{1}}(v_{0})$. By (21) and (23), we obtain that

[TABLE]

Consequently, we have

[TABLE]

for $n$ big enough, which contradicts (22). Therefore (20) holds.

For any $u\in Y_{k}$ and $t\in\Gamma_{\varepsilon_{1}}(u)$ with $\|u\|\geq\frac{r_{\infty}^{\xi}}{\varepsilon_{1}}$, it follows from $(W5)$ that

[TABLE]

We can choose $\rho_{k}>\max\left\{\frac{r_{\infty}^{\xi}}{\varepsilon_{1}},r_{k}\right\}$, then for any $u\in Y_{k}$ with $\|u\|=\rho_{k}$, it follows from (2), $(W6)$, (20) and (27) that

[TABLE]

which means $a_{k}(\lambda)\leq 0$ for $\rho_{k}$ large enough since $\xi\in(0,1)$.                                                                       $\Box$

By Lemma 3.1, we can obtain

Lemma 3.4**.**

There exist $\lambda_{n}\rightarrow 1$ as $n\rightarrow\infty$, $\{u_{n}(k)\}_{n=1}^{\infty}\subset E$ sastisfying $I_{\lambda_{n}}^{\prime}(u_{n}(k))=0$ and $I_{\lambda_{n}}(u_{n}(k))\in[\overline{b}_{k},\overline{c}_{k}]$, where $\overline{c}_{k}=\sup_{u\in B_{k}}I(u)$.

Proof of Theorem 1.2. From Lemmas 2.3 and 3.4, $\{u_{n}(k)\}_{n=1}^{\infty}$ is a bounded sequence. Similar to Theorem 1.1, we can conclude that $u_{n}(k)\rightarrow u(k)$ as $n\rightarrow\infty$, which implies that $I_{1}$ possesses a critical point $u_{k}$ with $I_{1}(u(k))\in[\overline{b}_{k},\overline{c}_{k}]$. Therefore, we obtain a sequence of critical points of $I_{1}$ since $\overline{b}_{k}\rightarrow+\infty$ as $k\rightarrow\infty$. $\Box$