On Homoclinic Solutions of a Semilinear -Laplacian Difference Equation with Periodic Coefficients

Abstract

We study the existence of homoclinic solutions for semilinear -Laplacian difference equations with periodic coefficients. The proof of the main result is based on Brezis-Nirenberg's Mountain Pass Theorem. Several examples and remarks are given.

1. Introduction

This paper is concerned with the study of the existence of homoclinic solutions for the -Laplacian difference equation

(1.1)

where is a sequence or real numbers, is the difference operator ,

(1.2)

is referred to as the -Laplacian difference operator, and functions and are -periodic in and satisfy suitable conditions.

In the theory of differential equations, a trajectory , which is asymptotic to a constant as is called doubly asymptotic or homoclinic orbit. The notion of homoclinic orbit is introduced by Poincaré [1] for continuous Hamiltonian systems.

Recently, there is a large literature on the use of variational methods to the existence of homoclinic or heteroclinic orbits of Hamiltonian systems; see [2–7] and the references therein.

In the recent paper of Li [8] a unified approach to the existence of homoclinic orbits for some classes of ODE's with periodic potentials is presented. It is based on the Brezis and Nirenberg's mountain-pass theorem [9]. In this paper we extend this approach to homoclinic orbits for discrete -Laplacian type equations.

Discrete boundary value problems have been intensively studied in the last decade. The studies of such kind of problems can be placed at the interface of certain mathematical fields, such as nonlinear differential equations and numerical analysis. On the other hand, they are strongly motivated by their applicability to mathematical physics and biology.

The variational approach to the study of various problems for difference equations has been recently applied in, among others, the papers of Agarwal et al. [10], Cabada et al. [11], Chen and Fang [12], Fang and Zhao [13], Jiang and Zhou [14], Ma and Guo [15], Mihăilescu et al. [16], Kristály et al. [17].

Along the paper, given two integer numbers , we will denote . Moreover, for every , we consider the following function

(1.3)

It is obvious that for all and . Moreover

(1.4)

Suppose that

(1.5)

(1.6)

Denote

(1.7)

Let us consider functions satisfying the following assumptions.

(F₁) The function is continuous in and -periodic in .

(F₂) The potential function of

(1.8)

satisfies the Rabinowitz's type condition:

There exist and such that

(1.9)

(F₃) as .

Further we consider the semilinear eigenvalue -Laplacian difference equation

(1.10)

where and we are looking for its homoclinic solutions, that is, solutions of (1.10) such that as .

In order to obtain homoclinic solutions of (1.10), we will use variational approach and Brezis-Nirenberg mountain pass theorem [9].

To this end, consider the functional , defined as

(1.11)

Our main result is the following.

Theorem 1.1.

Suppose that the function is positive and -periodic and the functions satisfy assumptions −. Then, for each , (1.10) has a nonzero homoclinic solution , which is a critical point of the functional .

Moreover, given a nontrivial solution of problem (1.10), there exist two integer numbers such that for all and , the sequence is strictly monotone.

The paper is organized as follows. In Section 2, we present the proof of the main result and discuss the optimality of the condition . In Section 3, we give some examples of equations modeled by this kind of problems and present some additional remarks.

2. Proof of the Main Result

Let be a sequence, and

(2.1)

It is well known that if , then . Indeed, if , there exists a positive integer number , such that for all satisfying it is verified that and, as consequence, and the series is convergent too.

Consider now the functional , defined as

(2.2)

with given in (1.7) and defined in (1.8).

We have the following result.

Lemma 2.1.

The functional is well defined, -differentiable, and its critical points are solutions of (1.10).

Proof.

By using the inequality for nonnegative and and

(2.3)

and the inclusion for , it follows that

(2.4)

Now, let us see that the series is convergent: by using , it follows that there exist and sufficiently large such that

(2.5)

Then, the series is convergent and the functional is well defined on .

It is Gâteaux differentiable and for :

(2.6)

and partial derivatives

(2.7)

are continuous functions.

Moreover the functional is continuously Fréchet-differentiable in . It is clear, by (2.7), that the critical points of are solutions of (1.10).

To obtain homoclinic solutions of (1.10) we will use mountain-pass theorem of Brezis and Nirenberg [9]. Recall its statement. Let be a Banach space with norm , and be a -functional. satisfies the  condition if every sequence of such that

(2.8)

has a convergent subsequence. A sequence such that (2.8) holds is referred to as -sequence.

Theorem 2.2 (mountain-pass theorem, Brezis and Nirenberg [9]).

Let be a Banach space with norm , and suppose that there exist , and such that

• (i) if ,

• (ii) .

Let , where

(2.9)

Then, there exists a sequence for . Moreover, if satisfies the condition, then is a critical value of , that is, there exists such that and .

Note that, by assumption (1.5), the norm in is equivalent to

(2.10)

Lemma 2.3.

Suppose that − hold, then there exist , and such that and

• (1) if ,

• (2) .

Proof.

By , there exists such that

(2.11)

Let ( defined in (1.6)), then, for , ,

(2.12)

which implies that for all .

Hence, by (2.11)

(2.13)

By , there exist , such that for all and .

Take , , if . Then, since

(2.14)

if is sufficiently large.

Then, we can take large enough, such that for , and (2.14) holds.

Lemma 2.4.

Suppose that the assumptions of Lemma 2.3 hold. Then, there exists and a -bounded sequence for .

Proof.

By Lemma 2.3 and Theorem 2.2 there exists a sequence such that

(2.15)

where

(2.16)

and is defined in the proof of Lemma 2.3.

We will prove that the sequence is bounded in . We have for

(2.17)

and, by ,

(2.18)

which implies that the sequence is bounded in .

Now we are in a position to prove Theorem 1.1.

Proof of Theorem 1.1.

For any , the sequence , given in Lemma 2.4, is bounded in and, in consequence, as . Let takes its maximum at . There exists a unique , such that and let . Then takes its maximum at . By the -periodicity of and , it follows that

(2.19)

Since is bounded in , there exists , such that weakly in . The weak convergence in implies that for every . Indeed, if we take a test function , , if , then

(2.20)

Moreover, for any

(2.21)

which implies that , which means that for every ,

(2.22)

Let us take with compact support, that is, there exist , such that if and if . The set of such elements is dense in because if and is such that if , if , then as . Taking in (2.22), due to the finite sums and the continuity of functions , we obtain, passing to a limit, that

(2.23)

From the density of in , we deduce that the previous equality is fulfilled for all and, in consequence, is a critical point of the functional , that is, is a solution of (1.10).

It remains to show that .

Assuming, on the contrary, that , we conclude that

(2.24)

By , for a given , there exists , such that if then, for every , the following inequalities holds:

(2.25)

By (2.24), for every , there exists a positive integer such that for all it follows that . Since the maximum value of is attained at , it follows that for and every

(2.26)

Then, by (2.25), for and every :

(2.27)

which implies that

(2.28)

Since is bounded in , and is arbitrary, by (2.28) we obtain a contradiction with . The proof of the first part is complete.

Now, let be a nonzero homoclinic solution of problem (1.10). Assume that it attains positive local maximums and/or negative local minimums at infinitely many points . In particular we can assume that . In consequence and .

From this, multiplying in (1.10) by , we have

(2.29)

By means of condition we arrive at the following contradiction:

(2.30)

Suppose now that function vanishes at infinitely many points . From condition we conclude that and, in consequence, . Therefore it has an unbounded sequence of positive local maximums and negative local minimums, in contradiction with the previous assertion.

As a direct consequence of the two previous properties, we deduce that, for large enough, function has constant sign and it is strictly monotone.

To illustrate the optimality of the obtained results, we present in the sequel an example in which it is pointed out that condition cannot be removed to deduce the existence result proved in Theorem 1.1.

Example 2.5.

Let be a -periodic sequence, , and be fixed. Consider problem (1.10) with

(2.31)

It is obvious that condition holds. Since we have that condition is trivially fulfilled. Concerning to condition , we have that

(2.32)

It is clear that for all and that for all if and only if .

When , the inequality holds if and only if either or and

(2.33)

As consequence, the inequality for all is satisfied if and only if , that is, condition does not hold.

Let us see that this problem has only the trivial solution for small values of the parameter .

Since , it is not difficult to verify that, for , the function is strictly decreasing for every integer . So, for in that situation, we have that

(2.34)

Suppose that there is a nontrivial solution of the considered problem, and moreover it takes some positive values. Let be such that . In such a case we deduce the following contradiction:

(2.35)

Analogously it can be verified that the solution has no negative values on .

3. Remarks and Examples

In this section we will consider some examples and remarks on applications and extensions of Theorem 1.1 to the existence of homoclinic solutions of difference equations of following types:

• (A) Second-order discrete -Laplacian equations of the form

  

  (3.1)

with.

• (B) Higher even-order difference equations. A model equation is the fourth-order extended Fisher-Kolmogorov equation

  

  (3.2)

with.

• (C) Second-order difference equations with cubic and quintic nonlinearities of the forms

  

  (3.3)

(3.4)

arising in mathematical physics and biology.

(A) Second-Order Discrete-Laplacian Equations.

The spectrum of the Dirichlet problem for (3.1), subject to Dirichlet boundary conditions

(3.5)

is studied in [17]. It is proved that if , and is a given function, then there exist two positive constants and with such that no is an eigenvalue of problem while any is an eigenvalue of problem . Moreover, we have

(3.6)

where and . Note that if is positive and then

(3.7)

where is a constant depending on , which implies that and as . It implies that for a given , there exists such that for any , the problem has a solution for every .

We extend this phenomenon, looking for homoclinic solutions of (3.1). Applying Theorem 1.1 with and , we obtain the following.

Corollary 3.1.

Suppose that the function is positive and -periodic and . Then, for each , (3.1) has a nonzero homoclinic solution.

Moreover, given a nontrivial solution of problem (3.1), there exist two integer numbers such that for all and , the sequence is strictly monotone.

(B) Higher Even-Order Difference Equations.

The statement of Theorem 1.1 can be extended to higher even-order difference equations. For simplicity we consider the fourth-order difference equations of the form

(3.8)

where for each , satisfy the assumptions −.

We consider the functional ,

(3.9)

where

(3.10)

which is well defined for ,.

Note that the series is convergent because

(3.11)

while is convergent since .

Now following the steps of the proof of Theorem 1.1 one can prove the following.

Theorem 3.2.

Suppose that , the function is positive and -periodic and the functions satisfy assumptions − and ,. Then, for each , (3.8) has a nonzero homoclinic solution , which is a critical point of the functional .

A typical example of (3.8) is (3.2), which is a discretization of a fourth-order extended Fisher-Kolmogorov equation. Homoclinic solutions for fourth-order ODEs are studied in [7] using variational approach and concentration-compactness arguments. As a consequence of Theorem 3.2 we obtain the following corollary.

Corollary 3.3.

Suppose that , the function is positive and -periodic and . Then, for each , (3.2) has a nonzero homoclinic solution .

(C) Second-Order Difference Equations with Cubic and Quintic Nonlinearities.

Our next example is (3.3), known as stationary Ginzubrg-Landau equation with cubic-quintic nonlinearity. We refer to [18, 19] and references therein. From physical point of view it is interesting the case ,,. Theorem 1.1 can be applied for with , -periodic, and positive. Then satisfies assumptions − with and as a consequence we have the following corollary.

Corollary 3.4.

Suppose that the functions , and are -periodic and and are positive. Then, for each , (3.3) has a nonzero homoclinic solution .

Moreover, given a nontrivial solution of problem (1.10), there exist two integer numbers such that for all and , the sequence is strictly monotone.

Moreover, we can prove that if in addition to conditions − the following condition holds:

(F₄) ,

the homoclinic solution of (1.10) is positive.

Indeed, let be a homoclinic solution of (1.10) and assume that holds. Suppose that there exists such that and let be such that . In consequence , which implies that

(3.12)

in contradiction with . Then for every .

If for some , we know that and, in consequence, , and we arrive at a contradiction as in the previous case.

We summarize above observations in the following.

Theorem 3.5.

Suppose that the function is positive and -periodic and the functions satisfy assumptions , , and . Then, for each , (1.10) has a nonzero homoclinic solution . If moreover holds, is a positive solution on that is strictly monotone for large enough.

In the case we can estimate the maximum of the solution , provided the additional assumption

(F₅) Assume that for all and function has the form , where is -periodic in , and for each , is increasing in for .

Let be the inverse function of for . We have that is increasing in for . Let be a positive homoclinic solution of (1.10) in view of last theorem and is its maximum. Note that, in view of the periodicity of coefficients, if is a solution of (1.10), then is also a solution of (1.10). Hence, we may assume that . Then and

(3.13)

and hence by properties of and

(3.14)

Thus

(3.15)

We summarize above observation in the following.

Corollary 3.6.

Let and suppose that the functions and satisfy assumptions of Theorem 3.5. Then, if in addition, satisfies condition , the positive homoclinic solution of the equation

(3.16)

satisfies the estimate (3.15).

Our next example, concerning Theorem 3.5, are (3.4) and

(3.17)

where .

Positive homoclinic solutions of corresponding differential equation are studied in [3] and periodic solutions in [20]. We suppose that the coefficients , , and are -periodic and there are constants ,,,, and such that

(3.18)

By Theorem 3.5, (3.17) has a positive solution , which is a critical point of the functional ,

(3.19)

Clearly, the positive solution of (3.17) is a positive solution of (3.4) too.

Further, let take its positive maximum at , then and, since , we have from (3.4) that

(3.20)

In view of (3.18), the last inequality implies

(3.21)

or

(3.22)

We obtain a positive lower bound for in the case

(3.23)

and (3.22) shows that blows up, that is, tends to as .

We summarize above facts in the following.

Corollary 3.7.

Let and and be -periodic sequences.

Assume that there are constants ,,,, and such that (3.18) holds. Then,

(3.24)

has a positive homoclinic solution and for ,

(3.25)

Let . By the last statement, if is the solution of the equation

(3.26)

then

(3.27)

Let be such that . Since is an infinite sequence of integers, by Dirichlet principle, there exists a fixed and a subsequence of , still denoted by , such that and . Note that if , then or .

Dedication

This work is dedicated to Professor Gheorghe Moroşanu on the occasion of his 60-th birthday.