Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential

where $n\in \mathbb{Z}$, $u\in {\mathbb{R}}^{\mathcal{N}}$, $\mathrm{△}u(n)=u(n+1)-u(n)$ is the forward difference operator, $p,L:\mathbb{Z}\to {\mathbb{R}}^{\mathcal{N}\times \mathcal{N}}$ and $W:\mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}}\to \mathbb{R}$. As usual, we say that a solution $u(n)$ of system (1.1) is homoclinic (to 0) if $u(n)\to 0$ as $n\to \pm \mathrm{\infty }$. In addition, if $u(n)\not\equiv 0$, then $u(n)$ is called a nontrivial homoclinic solution.

Moreover, system (1.1) does have its applicable setting as evidenced by monographs [1, 2]. System (1.2) can also be regarded as a special form of the Emden-Fowler equation appearing in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting system, and many well-known results concerning properties of solutions of (1.2) are collected in [3].

or other superquadratic growth conditions, where and in the sequel, $(\cdot ,\cdot )$ denotes the standard inner product in ${\mathbb{R}}^{\mathcal{N}}$ and $|\cdot |$ is the induced norm.

When $W(n,x)$ is of subquadratic growth at infinity, Tang and Lin [9] recently established the following results on the existence of homoclinic solutions of system (1.1).

Theorem A [9]

where $\phi (s)=O(s^{{\gamma }_1-1})$ as $s\to 0^{+}$;

Then system (1.1) possesses at least one nontrivial homoclinic solution.

Theorem B [9]

Assume that $p(n)$ is an $\mathcal{N}\times \mathcal{N}$ real symmetric positive definite matrix for all $n\in \mathbb{Z}$, and that L and W satisfy (L), (W1), (W2) and the following assumptions:

(W5) $W(n,-x)=W(n,x)$, $\mathrm{\forall }(n,x)\in \mathbb{Z}\times {\mathbb{R}}^{\mathcal{N}}$.

Then system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions.

When $L(n)$ satisfies (L) in Theorems A and B, assumption (W1) is optimal in some sense, essentially, the summable functions $a_1,a_2\in l^{2/(2-{\gamma }_1)}(\mathbb{R},[0,+\mathrm{\infty }))$ are necessary; see Lemma 2.2 in Section 2.

Now a natural question is whether the conditions on the potential $W(n,x)$ can be further relaxed when one imposes stronger conditions on $L(n)$?

In the present paper, we give a positive answer to the above question. In fact, we employ Clark’s theorem in critical point theory to establish new existence criteria to guarantee that system (1.1) has at least one or multiple homoclinic solutions under the following assumption instead of (L):

Our main results are the following two theorems.

Theorem 1.1 Assume that $p(n)$ is an $\mathcal{N}\times \mathcal{N}$ real symmetric positive definite matrix for all $n\in \mathbb{Z}$, that L satisfies (L _ν ) and W satisfies the following assumptions:

where $\phi (s)=O(s^{{\gamma }_3-1})$ as $s\to 0^{+}$, $max\{1,2/(3-\nu )\}<{\gamma }_3<2$;

Then system (1.1) possesses at least one nontrivial homoclinic solution.

Theorem 1.2 Assume that $p(n)$ is an $\mathcal{N}\times \mathcal{N}$ real symmetric positive definite matrix for all $n\in \mathbb{Z}$, and that L and W satisfy (L _ν ), (W1′), (W2′), (W5) and the following assumption:

Then system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions.

Remark 1.3 Obviously, assumptions (W1′), (W2′), (W3′) and (W4′) are weaker than (W1), (W2), (W3) and (W4), respectively.

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proofs of our theorems. In Section 4, we give some examples to illustrate our results.

respectively.

Lemma 2.1 [9]

where $\alpha =inf\{(p(n)x,x):n\in \mathbb{Z},x\in {\mathbb{R}}^{\mathcal{N}},|x|=1\}$ and $\beta =inf\{(L(n)x,x):n\in \mathbb{Z},x\in {\mathbb{R}}^{\mathcal{N}},|x|=1\}$.

This shows that (2.4) holds.

This shows that $\{u_k\}$ possesses a convergent subsequence in $l^q(\mathbb{Z},{\mathbb{R}}^{\mathcal{N}})$. Therefore, E is compactly embedded in $l^q(\mathbb{Z},{\mathbb{R}}^{\mathcal{N}})$ for $1\le q\in (2/(3-\nu ),2)$. □

This shows (2.12) holds. □

Furthermore, the critical points of f in E are solutions of (1.1) with $u(\pm \mathrm{\infty })=0$.

So, the critical points of f in E are the solutions of system (1.1) with $u(\pm \mathrm{\infty })=0$.

which implies the continuity of $f^{\prime }$. The proof is complete. □

Lemma 2.5 [10]

Let E be a real Banach space and let $f\in C^1(E,\mathbb{R})$ satisfy the (PS)-condition. If f is bounded from below, then $c={inf}_{E}f$ is a critical value of f.

Lemma 2.6 [11]

Let E be a real Banach space, let $f\in C^1(E,\mathbb{R})$ with f even, bounded from below, and satisfying the (PS)-condition. Suppose that $f(0)=0$, there is a set $K\subset E$ such that K is homeomorphic to $S^{k-1}$ by an odd map, and ${sup}_{K}f<0$. Then f possesses at least k distinct pairs of critical points.

Since $1<{\gamma }_1<{\gamma }_2<2$, (3.2) implies that $f(u)\to +\mathrm{\infty }$ as $\parallel u\parallel \to +\mathrm{\infty }$. Consequently, f is bounded from below.

Since $〈f^{\prime }(u_k)-f^{\prime }(u_0),u_k-u_0〉\to 0$, it follows from (3.10) and (3.11) that $u_k\to u_0$ in E. Hence, f satisfies the (PS)-condition.

By Lemma 2.5, $c={inf}_{E}f(u)$ is a critical value of f, that is, there exists a critical point $u^{\ast }\in E$ such that $f(u^{\ast })=c$.