The purpose of this paper is to study a class of discrete nonlinear Schrödinger equations. Under a weak superlinearity condition at infinity instead of the classical Ambrosetti-Rabinowitz condition, the existence of standing waves of the equations is obtained by using the Nehari manifold approach.

The discrete nonlinear Schrödinger (DNLS) equation was first derived in the context of nonlinear optics by Christodoulides and Joseph [

For the analytical study, many authors studied the existence results of standing wave solutions for DNLS equations. Much of the works concerns the periodic DNLS equations [

In this paper, we consider higher-dimensional generalizations of DNLS equation

We assume that the nonlinearity

where

We are concerned with the existence of ground state solutions, that is, solutions corresponding to the least positive critical value of the variational functional. To obtain the existence of ground states, usually besides the growth condition on the nonlinearity and a Nehari type condition, the following classical Ambrosetti-Rabinowitz superlinear condition (see, e.g., [

In this paper, instead of (

This paper is organized as follows. In Section

In order to apply the critical point theory, we will establish the corresponding variational framework associated with (

For some positive integer

Let

Define the space

Now we consider the variational functional

The following lemma plays an important role in this paper; it was established in [

If

for any

the spectrum

By Lemma

Now we are ready to state the main results.

Suppose that conditions (

If

If

If

In [

In [

In [

Since

To prove the multiplicity results, we need the following lemma.

Let

In this section, we always assume that

We define the Nehari manifold

To prove the main results, we need some lemmas.

Suppose that conditions (

(1) From

(2) For all

Suppose that conditions (

(1) and (2) are easy to be shown from

Since

Under the assumptions (

Let

By (1) and (3) of Lemma

For each compact subset

Suppose that, by contradiction,

(1) The mapping

(2) The mapping

(1) Suppose that

Let

Firstly, we prove that

Suppose that

According to Lemma

Finally, we show that there exists a convergent subsequence of

The first term

By

The proof is complete.

Now we define the functional

(1)

(2)

(3)

(4)

(1) Let

(2) follows from (1). Note only that since

(3) Let

(4) By (

(1) If

(2) If

Let

Let

(3) If

This completes Theorem

This work is supported by Program for the National Natural Science Foundation of China (no. 11071283) and Yuncheng University Science Foundation (nos. JY-2011026, JY-2011038, JY-2011039, and JC-2009024).