We consider the existence of solutions to boundary value problems for the discrete generalized Emden-Fowler equation. By means of the minimax methods in the critical point theory, some new results are obtained. Two examples are also given to illustrate the main results.

Let

In this paper, we consider the following boundary value problem (BVP for short) consisting of the discrete generalized Emden-Fowler equation:

Equation (

Recently, Yu and Guo in [

Before giving the main results, we first set

Suppose that

There are constants

One has

either

The matrix

There are functions _{1}), (F_{2})(i), and (P_{1}) are satisfied. Then the BVP (

Suppose that

For any

The matrix

Suppose that _{1}) and (P_{2}). Then the BVP (

Since for all

Suppose that

For any

One has

either

The matrix

There are functions _{4}), (F_{5})(i) and (P_{3}) are satisfied. Then the BVP (

Assume that (F_{4}) holds. If one of the following conditions is satisfied: (_{5})(i) holds, (_{5})(ii) holds, then, the BVP (

Let

Define the functional

When the matrix

Set

We will make use of the least action principle and saddle point theorem to obtain the critical points of

Let

Let

there exist constants

there exist

As shown in [

In order to prove Theorem

Assume that conditions (F_{1}), (F_{2}), and (P_{3}) hold. Then the functional

First suppose that (F_{1}), (F_{2})(i), and (P_{3}) hold. Recall that

Since _{1}), (

Take

By (_{2})(i) imply that

Now, suppose that (F_{1}), (F_{2})(ii), and (P_{3}) hold. By a similar argument as above, we know also that

Assume that (F_{1}), (F_{2})(i), and (P_{1}) hold. The proof for the case when (F_{1}), (F_{2})(ii), and (P_{1}) hold is similar and will be omitted here. Since

Suppose that the matrix _{1}), (_{2})(i), we have

On the other hand, for any _{1}), (

Let

Suppose that

Suppose that

For any _{2})(i) and

Since

Since the matrix

Suppose that _{3}) that there exists a positive constant

We now prove that the functional

For any

Let

Suppose that

Suppose that

Since

This is immediate from Theorem

The following lemma is useful for proving Theorem

Under the condition (F_{5}), the functional

First suppose that (F_{5})(i) holds. Let _{5})(i), we have
_{5})(i) implies that there exists a constant

Now, suppose that (F_{5})(ii) holds. By a similar argument as above, we know also that

Assume that (F_{4}), (F_{5})(i), and (P_{3}) hold. The proof for the case when (F_{4}), (F_{5})(ii), and (P_{3}) hold is similar and will be omitted here. Due to (P_{3}), _{5})(i), we can obtain that for any given _{4}) that

Now we prove

On the other hand, by (F_{4}), there exists a positive constant

It follows from (

The proof of Corollary

The authors thank the referees for their valuable comment and careful corrections. This project is supported by Science and Technology Plan Foundation of Guangzhou (No. 2006J1-C0341).