Multiple homoclinic solutions for a class of nonhomogeneous Hamiltonian systems

By introducing a new superquadratic condition, we obtain the existence of two nontrivial homoclinic solutions for a class of perturbed second order Hamiltonian systems which are obtained by the mountain pass theorem and Ekeland’s variational principle.

In this paper, we consider the existence of two nontrivial homoclinic solutions for the following second order Hamiltonian systems:

for all \(t \in R\), where \(W\in C^{1}( R\times R^{N}, R)\), L: \(R\rightarrow R^{N^{2}}\) is a matrix-valued function and \(f\in C( R, R^{N})\). A solution \(u(t)\) of problem (1) is homoclinic (to 0) if \(u(t)\rightarrow 0\) as \(t\rightarrow \pm \infty \). Moreover, if \(u(t)\neq 0\), \(u(t)\) is called a nontrivial homoclinic solution. Here and subsequently, \(\nabla W(t,x)\) denotes the gradient with respect to the x variable.

The homoclinic solutions have been proved to be important in studying the behavior of dynamic systems. There have been many papers concerning this topic by using the variational methods since the remarkable results by Ambrosetti and Rabinowitz [1]. Because of the lack of compactness, this problem is more difficult than studying the existence of periodic solutions. In order to get the compactness of embedding theorem back, many conditions have been proposed (see [1–41]). Two kinds of important conditions are periodic and coercive conditions. The periodic condition was introduced by Rabinowitz [19] in 1990 to discuss the existence of homoclinic solutions for problem (1) as the limit of a sequence of subharmonics which are obtained by the mountain pass theorem. The following coercive condition is a classical condition introduced by Rabinowitz and his co-author [20].

\((L')\) :

\(L\in C( R, R^{N^{2}})\) is a symmetric and positively definite matrix for all \(t\in R\) and

$$\begin{aligned} \inf_{\vert x\vert =1}\bigl(L(t)x,x\bigr)\rightarrow +\infty \quad \mbox{as } \vert t\vert \rightarrow \infty. \end{aligned}$$

Condition \((L')\) has been studied by many other mathematicians to deal with the nonperiodic systems. After then, there have been some other coercive conditions introduced by other mathematicians.

By using the variational methods to study problem (1), the growth conditions of \(W(t,x)\) are needed. These conditions are mainly classified into three cases: the superquadratic case, the subquadratic case, and the asymptotically quadratic case. In this paper, we mainly consider the superquadratic case. The following growth condition is a classical superquadratic condition known as the \((AR)\) condition.

\((AR)\) :

there exists a constant \(\theta > 2\) such that

$$\begin{aligned} 0 < \theta W(t,x) \leq \bigl(\nabla W(t,x),x\bigr) \end{aligned}$$

However, the \((AR)\) condition is so strong that many functions cannot be involved. In order to study problem (1) with different potentials, many other superquadratic conditions are proposed. In 2009, Ding and Lee [8] introduced the following generalized superquadratic condition.

\((GS)\) :

There exist \(\epsilon \in (0,1)\) and \(r_{1}\), \(d_{0}>0\) such that

$$\begin{aligned} \widetilde{W}(t,x)\geq d_{0}\frac{(\nabla W(t,x),x)}{\vert x\vert ^{2-\epsilon }}\quad \mbox{for all } t\in R \mbox{ and } \vert x\vert \geq r_{1}, \end{aligned}$$

Some examples are given to show the difference between \((GS)\) and \((AR)\) conditions. The following superquadratic condition is used by Lv and Tang [14] to obtain infinitely many homoclinic solutions for problem (1) when \(W(t,x)\) is even in x.

\((MC)\) :

There exists \(\varsigma \geq 1\) such that

$$\begin{aligned} \varsigma \widetilde{W}(t,x)\geq \widetilde{W}(t,\varsigma x) \end{aligned}$$

\((SQ)\) :

\(\frac{\widetilde{W}(t,x)}{W(t,x)}\vert x\vert ^{2}\rightarrow +\infty\) as \(\vert x\vert \rightarrow \infty \) uniformly in \(t\in R\), where

$$\begin{aligned} \widetilde{W}(t,x)=\bigl(\nabla W(t,x),x\bigr)-2W(t,x). \end{aligned}$$

In 2018, Wu et al. [32] showed the existence of homoclinic solutions for problem (1) without periodic or even conditions. In 2015, Xu et al. [36] showed the existence of two solutions for problem (1) with a nonzero perturbation. In the same year, Zhang and Yuan [40] obtained two homoclinic solutions for a class of perturbed Hamiltonian systems under the \((AR)\) condition. In this paper, we introduce a new superquadratic condition to study problem (1) with small forcing terms. The following theorem is our main result.

Theorem 1.1

Suppose that W and L satisfy the following conditions:

\((L)\) :

For the smallest eigenvalue of \(L(t)\), i.e., \(l(t)\equiv \inf_{\vert x\vert =1} L(t)x\cdot x\), there exists a constant \(\nu <1\) such that \(l(t)\vert t\vert ^{\nu -2}\rightarrow +\infty \) as \(\vert t\vert \rightarrow \infty \).

\((W1)\) :

\(W(t,0)=0\) for all \(t \in R\) and \(\nabla W(t,x)=o(\vert x\vert )\) as \(\vert x\vert \rightarrow 0\) uniformly for \(t\in R\);

\((W2)\) :

\((\nabla W(t,x),x)\geq 2W(t,x)\geq 0\) for all \((t,x) \in R \times R^{N}\);

\((W3)\) :

\(W(t,x) / \vert x\vert ^{2} \rightarrow +\infty \) as \(\vert x\vert \rightarrow \infty \) uniformly in t;

\((W4)\) :

There exist \(\tau >2\) and \(d_{1}>0\) such that

$$\begin{aligned} \bigl\vert \nabla W(t,x)\bigr\vert \leq d_{1}\bigl(1+\vert x \vert ^{\tau -1}\bigr)\quad \textit{for all } t\in R; \end{aligned}$$

\((W5)\) :

There exist constants \(\mu \geq 1\), \(\lambda_{0}\in (0,1)\), \(d_{2}>0\), and \(r_{\infty }>0\) such that

$$\begin{aligned} \biggl( \frac{1-\lambda^{2}}{2}-\lambda \biggr) \bigl(\nabla W(t,x),x\bigr)+W(t, \lambda x)-W(t,x)\geq -d_{2}\lambda^{\mu }\vert x\vert ^{\mu } \end{aligned}$$

Then there exists \(\delta >0\) such that, for any \(f\not \equiv 0\) satisfying

system (1) possesses at least two nontrivial homoclinic solutions.

Remark 1

In Theorem 1.1, the perturbation f is not required to be integrable.

Remark 2

Consider the following example:

where \(s>2\) and \(\epsilon \in (0,s-2)\). It is easy to check that (3) satisfies the conditions of Theorem 1.1 and but not the \((AR)\) condition. As we know, Theorem 1.1 is the first result to obtain the existence of two homoclinic solutions for problem (1) without the \((AR)\) condition.

Let A be a self-adjoint extension of the operator \(-(d^{2}/dt^{2})+L(t)\) with the domain \(\mathcal{D}(A)\subset L^{2}( R, R^{N})\). Let \(E=\mathcal{D}(\vert A\vert ^{1/2})\) be the domain of \(\vert A\vert ^{1/2}\) and define on E the inner product and the norm as \((u,w)_{0}=(\vert A\vert ^{1/2}u,\vert A\vert ^{1/2}w)_{2}+(u,w)_{2}\) and \(\Vert u\Vert _{0}=(u,u)^{1/2}\), respectively, where \((\cdot,\cdot)_{2}\) denotes the inner product of \(L^{2}\). Then E is a Hilbert space.

It is known that the spectrum \(\sigma (A)\) consists of eigenvalues numbered in \(\lambda_{1}\leq \lambda_{2} \leq \cdots \rightarrow \infty \), and a corresponding system of eigenfunctions \((e_{n})(Ae _{n} =\lambda_{n}e_{n})\) forms an orthogonal basis in \(L^{2}\). Let \(n^{-}=\#\{i\vert \lambda_{i}<0\}\), \(n^{0}=\#\{i\vert \lambda_{i}=0\}\), and \(\bar{n}=n^{-}+n^{0}\). Set \(E^{-}=\operatorname{span}\{e_{1},\ldots,e_{n^{-}}\}\), \(E^{0}=\operatorname{span}\{e_{n^{-}+1},\ldots,e_{\bar{n}}\}\), and \(E^{+}=\overline{\operatorname{span}\{e_{\bar{n}+1},\ldots \}}\). Then \(E=E^{-}\oplus E^{0}\oplus E ^{+}\). The inner product and the norm on E are introduced as

where \(u=u^{-}+u^{0}+u^{+}\) and \(w=w^{-}+w^{0}+w^{+}\in E=E^{-}\oplus E^{0}\oplus E^{+}\). Furthermore, let \(I: E \rightarrow R\) be the functional defined by

It is known that the critical points of I in E are the homoclinic solutions of (1). One can easily check that \(I\in C^{1}(E, R)\) and

By Lemma 2.2 in [6] we can conclude that E is compactly embedded in \(L^{p}\) for any \(p\in [1,+\infty ]\), which implies that there exists a constant \(C_{p}>0\) such that

Lemma 2.1

Suppose that the conditions of Theorem 1.1 hold, then there exist constants α, \(\varrho >0\) such that \(I\vert_{S} \geq \alpha \), where \(S=\{u\in E\vert \Vert u\Vert =\varrho \}\).

Proof

By \((W1)\), for any \(\varepsilon >0\), there exists \(\sigma >0\) such that

Then it follows from \((W1)\) and \((W2)\) that

for all \(t\in R\) and \(\vert x\vert \leq \sigma \). Let \(\varepsilon_{0}=\frac{1}{4C _{2}}\), then there exists \(\sigma_{0}>0\) such that (7) holds for all \(t\in R\) and \(\vert x\vert \leq \sigma_{0}\). Set

which implies \(0 < \Vert u\Vert _{L^{\infty }} \leq \sigma_{0}\) for all \(u \in S\). Then it follows from the definition of I, (7) and (2) that

By the definitions of ϱ and α, there exists \(\delta_{0}>0\) such that \(I\mid_{S} \geq \alpha \) for any f satisfying (2). □

Lemma 2.2

Suppose that the conditions of Theorem 1.1 hold, then there is \(e\in E\) such that \(\Vert e\Vert > \varrho \) and \(I(e) \leq 0\), where ϱ is defined in Lemma 2.1.

Proof

It follows from \((W3)\) that there exist \(T>0\), \(\xi >0\), and \(\varepsilon_{1}>0\) such that

for all \(t\in [-T,T]\) and \(\vert x\vert > \xi \). Set \(\zeta =\max \{\vert W(t,x)\vert \vert t \in [-T,T], \vert x\vert \leq \xi \}\), hence we have

Set

where \(\omega =\frac{\pi }{T}\), \(e=(1,0,\ldots,0)\). It can be easily checked that \(( \frac{\pi^{2}}{2T^{2}}+\varepsilon_{1} ) m>M\), where

By (4), for every \(r \in R\setminus \{0\}\), the following inequality holds:

which implies that there exists \(r \in R \setminus \{ 0\} \) such that \(\Vert r Q_{1}\Vert > \varrho \) and \(I(r Q_{1}) < 0\). Set \(e(t)=r Q_{1}(t)\). Then \(\Vert e\Vert > \varrho \) and \(I(e) < 0\). □

Lemma 2.3

Suppose that the conditions of Theorem 1.1 hold, then I satisfies the \((C)\) condition.

Proof

Assume that \(\{u_{n}\} \subset E\) is a sequence such that \(\{I(u_{n})\}\) is bounded and \(\Vert I^{\prime }(u_{n})\Vert (1+\Vert u_{n}\Vert ) \rightarrow 0\) as \(n \rightarrow \infty \). Then there exists a constant \(M_{1}> 0\) such that

Now we prove that \(\{u_{n}\}\) is bounded in E. Arguing in an indirect way, we assume that \(\Vert u_{n}\Vert \rightarrow +\infty \) as \(n\rightarrow \infty \). Set \(z_{n}=\frac{u_{n}}{\Vert u_{n}\Vert }\), then \(\Vert z_{n}\Vert =1\), which implies that there exists a subsequence of \(\{z_{n}\}\), still denoted by \(\{z_{n}\}\), such that \(z_{n}\rightharpoonup z_{0}\) in E. By (4) and (8), we get

which implies that

for n large enough. The following discussion is divided into two cases.

Case 1: \(z_{0}\equiv 0\). Let \(s>\frac{\delta C_{1}}{2}\). From \((W1)\) and \((W4)\), we can deduce that there exists \(M_{2}>0\) such that

and

By the compactness of the embedding, one can obtain

Set \(\lambda_{n}=\frac{s}{\Vert u_{n}\Vert }\). It follows from (8)–(13), \((W2)\), and \((W5)\) that

for s and n large enough, which is a contradiction. Hence \(\Vert u_{n}\Vert \) is still bounded in this case, which implies that \(\{u_{n}\}\) is bounded in E.

Case 2: \(z_{0}\not \equiv 0\). Let \(\Omega =\{t\in R\vert \vert z_{0}(t)\vert >0 \}\). Then we can see that \(\operatorname{meas}(\Omega)>0\), where meas denotes the Lebesgue measure. Since \(\Vert u_{n}\Vert \rightarrow +\infty \) as \(n\rightarrow \infty \) and \(\vert u_{n}(t)\vert =\vert z_{n}(t)\vert \cdot \Vert u_{n}\Vert \), then we have \(\vert u_{n}(t)\vert \rightarrow +\infty \) as \(n\rightarrow \infty \) for a.e. \(t\in \Omega \). By \((W2)\), \((W3)\), and Fatou’s lemma, we can obtain

which contradicts (10). So \(\Vert u_{n}\Vert \) is bounded in this case. □

By a standard argument, we see that \(\{u_{n}\}\) has a convergent subsequence in E. Hence I satisfies the \((C)\) condition.

Proof of Theorem 1.1

The proof of this theorem is divided into two steps.

Step 1: We show that there exists a function \(u_{0}\in E\) such that \(I'(u_{0})=0\) and \(I(u_{0})<0\). Let \(f(t)=(f_{1}(t), f_{2}(t),\ldots,f_{N}(t))\), where \(f_{i}(t)\in C( R, R)\) (\(i=1,2,\ldots,N\)). Since \(f\not \equiv 0\), there exists \(i_{0}\in [1,N]\cap Z\) such that \(f_{i_{0}}(t)\not \equiv 0\). Without loss of generalization, we assume that there exist an interval \((a,b)\subset R\) and a constant \(A>0\) such that

We choose a function \(\psi_{0}\in C_{0}^{\infty }(a,b)\) satisfying

Set \(\psi (t)=(\psi_{1}(t),\psi_{2}(t),\ldots,\psi_{N}(t))\), where \(\psi_{j}(t)=0\) for all \(j\in [1,N]\cap Z\setminus \{i_{0}\}\) and \(\psi_{i}(t)=\psi_{0}(t)\) for \(j=i_{0}\). Therefore, \(\psi \in E\) and we can deduce that

Hence we have

for \(r>0\) small enough. Then we obtain

where ϱ is defined in Lemma 2.1 and \(B_{\varrho }=\{u\in E\vert \Vert u\Vert \leq \varrho \}\). By Ekeland’s variational principle, there exists a sequence \(\{u_{n}\}\subset B_{\varrho }\) such that

and

for all \(w\in B_{\varrho }\). Then, by a standard procedure, we can show that \(\{u_{n}\}\) is a \((C)\) sequence of I. Therefore, it follows from Lemma 2.3 that there exists a function \(u_{0}\in E\) such that \(I'(u_{0})=0\) and \(I(u_{0})<0\).

Step 2: By the mountain pass theorem and Lemmas 2.1–2.3, there exists \({\widetilde{u}_{0}}\in E\) such that \(I'(\widetilde{u}_{0})=0\) and \(I(\widetilde{u}_{0})>0\).

Then we finish the proof of Theorem 1.1. □

The authors would like to thank the editor and the referees for their valuable comments and suggestions, which improved the quality of our paper.

Funding

The first author is supported by the Scientific Research Foundation of Huaiyin Institute of Technology (15HGZ014, 17HGZ004). The second author is supported by the National Natural Science Foundation of China (No. 11626198) and the Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No. 2017CXTD02) and the Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No. 18TD0013).

Affiliations

1. Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai’an, China

   • Chunhua Deng

2. College of Science, Southwest Petroleum University, Chengdu, China

   • Dong-Lun Wu

3. Institute of Nonlinear Dynamics, Southwest Petroleum University, Chengdu, China

   • Dong-Lun Wu

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Chunhua Deng.

Competing interests

The authors declare that they have no competing interests.

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