Three Solutions for a Discrete Nonlinear Neumann Problem Involving the -Laplacian

Abstract

We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving the -Laplacian. Our approach is based on three critical points theorems.

1. Introduction

In these last years, the study of discrete problems subject to various boundary value conditions has been widely approached by using different abstract methods as fixed point theorems, lower and upper solutions, and Brower degree (see, e.g., [1–3] and the reference given therein). Recently, also the critical point theory has aroused the attention of many authors in the study of these problems [4–12].

The main aim of this paper is to investigate different sets of assumptions which guarantee the existence and multiplicity of solutions for the following nonlinear Neumann boundary value problem

where is a fixed positive integer, is the discrete interval , for all , is a positive real parameter, , , is the forward difference operator, , , and is a continuous function.

In particular, for every lying in a suitable interval of parameters, at least three solutions are obtained under mutually independent conditions. First, we require that the primitive of is sublinear at infinity and satisfies appropriate local growth condition (Theorem 3.1). Next, we obtain at least three positive solutions uniformly bounded with respect to , under a suitable sign hypothesis on , an appropriate growth conditions on in a bounded interval, and without assuming asymptotic condition at infinity on (Theorem 3.4, Corollary 3.6). Moreover, the existence of at least two nontrivial solutions for problem () is obtained assuming that is sublinear at zero and superlinear at infinity (Theorem 3.5).

It is worth noticing that it is the first time that this type of results are obtained for discrete problem with Neumann boundary conditions; instead of Dirichlet problem, similar results have been already given in [6, 9, 13]. Moreover, in [14], the existence of multiple solutions to problem () is obtained assuming different hypotheses with respect to our assumptions (see Remark 3.7).

Investigation on the relation between continuous and discrete problems are available in the papers [15, 16]. General references on difference equations and their applications in different fields of research are given in [17, 18]. While for an overview on variational methods, we refer the reader to the comprehensive monograph [19].

2. Critical Point Theorems and Variational Framework

Let be a real Banach space, let be two functions of class on , and let be a positive real parameter. In order to study problem (), our main tools are critical points theorems for functional of type which insure the existence at least three critical points for every belonging to well-defined open intervals. These theorems have been obtained, respectively, in [6, 20, 21].

Theorem 2.1 (see [11, Theorem 2.6]).

Let be a reflexive real Banach space, be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on , be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that

(2.1)

Assume that there exist and , with such that

()  ,

()  for each the functional is coercive.

Then, for each , the functional has at least three distinct critical points in .

Theorem 2.2 (see [7, Corollary 3.1]).

Let be a reflexive real Banach space, be a convex, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on , and let be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that

(2.2)

Assume that there exist two positive constants , and , with such that

(b₁),

(b₂),

(b₃) for each , and for every , which are local minima for the functional such that and , and one has .

Then, for each , the functional admits at least three critical points which lie in .

Finally, for all , we put

(2.3)

where we read if this case occurs.

Theorem 2.3 (see [8, Theorem 2.3]).

Let be a finite dimensional real Banach space. Assume that for each one has

(e).

Then, for each , the functional admits at least three distinct critical points.

Remark 2.4.

It is worth noticing that whenever is a finite dimensional Banach space, a careful reading of the proofs of Theorems 2.1 and 2.2 shows that regarding to the regularity of the derivative of and , it is enough to require only that and are two continuous functionals on .

Now, consider the -dimensional normed space endowed with the norm

(2.4)

In the sequel, we will use the following inequality:

(2.5)

Moreover, put

(2.6)

where for every . It is easy to show that and are two functionals on .

Next lemma describes the variational structure of problem (), for the reader convenience we give a sketch of the proof, see also [14],

Lemma 2.5.

is a Banach space. Let , be a solution of problem () if and only if is a critical point of the functional .

Proof.

Bearing in mind both that a finite dimensional normed space is a Banach space and the following partial sum:

(2.7)

for every and , standard variational arguments complete the proof.

Finally, we point out the following strong maximum principle for problem ().

Lemma 2.6.

Fix such that

(2.8)

Then, either in , or .

Proof.

Let be such that . An immediate computation gives

(2.9)

From this, by (2.8), we obtain

(2.10)

so , that is . Moreover, assuming that , from the preciding inequality and nonnegativity of , one has

(2.11)

so . Thus, repeating these arguments, the conclusion follows at once.

3. Main Results

For each positive constants and , we write

(3.1)

Now, we give the main results.

Theorem 3.1.

Assume that there exist three positive constants , , and with , and such that

(i₁),

(i₂).

Then, for every

(3.2)

problem (P_λ^f) admits at least three solutions.

Proof.

We apply Theorem 2.1, by putting and defined as in (2.6) on the space . An easy computation ensures the regularity assumptions required on and ; see Remark 2.4. Therefore, it remains to verify assumptions () and (). To this hand, we put

(3.3)

and we pick , defined by putting

(3.4)

Clearly, since , one has , and in addition, by (2.5), we have

(3.5)

On the other hand, we compute

(3.6)

Therefore, by (), combining (3.5) and (3.6), it is clear that () holds. Moreover, one has

(3.7)

Now, fix as in the conclusion; first, we observe that for every , one has

(3.8)

Next, by (), there exist two positive constants and such that

(3.9)

Hence, for every , we get

(3.10)

At this point, since , it is clear that the functional turns out to be coercive.

Remark 3.2.

We note that hypothesis () can be replaced with the following:

(i₂¹)  .

Arguing as before, there exist two constant and such that

(3.11)

Hence, for every , it easy to see that

(3.12)

with .

Remark 3.3.

It is worth noticing that a careful reading of the proof of Theorem 3.1 shows that, provided that and under the only condition (), problem () admits at least one solution for every and at least three solutions for every , whenever there exists for which .

Theorem 3.4.

Let be a continuous function in such that for some . Assume that there exist three positive constants , , and with such that

() for each ,

().

Then, for each , problem () admits at least three positive solutions , , such that

(3.13)

for all , .

Proof.

Consider the auxiliary problem

where is a continuous function defined putting

(3.14)

From (), owing to Lemma 2.6, any solution of problem () is positive. In addition, if it satisfies also the condition , and for every , clearly it turns to be also a positive solution of (). Therefore, for our goal, it is enough to show that our conclusion holds for (). In this connection, our aim is to apply Theorem 2.2. Fix in and let , and as before. Now, take

(3.15)

From (2.5), arguing as before, we obtain

(3.16)

for all such that , and

(3.17)

for all such that .

Therefore, one has

(3.18)

as well as

(3.19)

On the other hand, pick , defined as in (3.4), bearing in mind (3.6), and from , we obtain Moreover, taking into account (3.18), (3.19), from (), assumptions () and () follow. Further, again from (3.18), (3.19), and (3.6), one has that

(3.20)

Now, let and be two local minima for such that and . Owing to Lemmas 2.5 and 2.6, they are two positive solutions for () so , for all and for all . Hence, since one has for all , is verified.

Therefore, the functional admits at least three critical points , , which are three positive solutions of (). Finally, from (2.5), for , one has

(3.21)

and the proof is completed.

Theorem 3.5.

Let be a continuous function such that for some . Assume that there exist four constants , , , and , with , and such that

(), for all .

Then, for each , where

(3.22)

problem () admits at least three nontrivial solutions.

Proof.

Our aim is to apply Theorem 2.3 with and as above. Fix , and there is such that . Setting and arguing as in the proof of Theorem 3.1, one has

(3.23)

that is . Moreover, denote

(3.24)

it is a simple matter to show that for each , one has

(3.25)

Hence, from (), for each , we get

(3.26)

Therefore, since and , condition () is verified. Hence, from Theorem 2.3, the functional admits three critical points, which are three solutions for (). Since for some , they are nontrivial solutions, and the conclusion is proved.

Corollary 3.6.

Let be a continuous function such that for some . Assume that there exist four constants , , , and with and such that

(l₁),

(l₂), for all .

Then, for every

(3.27)

problem () admits at least three solutions.

Proof.

Our claim is to prove that condition () of Theorem 2.3 holds for every . Indeed, from (), arguing as in (3.23), one has that . Moreover, by (), from (3.26) with , for every , we have

(3.28)

where , which implies condition ().

Remark 3.7.

In [14], by Mountain Pass Theorem, the authors established the existence of at least one solution for problem () requiring the following conditions:

(θ₁) for uniformly in ,

(θ₂) there exist two positive constants and with such that

(3.29)

for every and .

Moreover, they remember that the above conditions imply, respectively, the following:

(θ₃) for uniformly in ,

(θ₄) there exist two positive constants and such that

(3.30)

Next result shows that under more general conditions than () and (), problem () has at least two nontrivial solutions.

Theorem 3.8.

Assume that () holds and in addition

(θ₅).

Then, problem () has at least two nontrivial solutions.

Proof.

We claim that the functional admits a local minimum at zero and a local nonzero maximum. To this end, we observe that by (), there exist and such that

(3.31)

Hence, bearing in mind Lemma 2.5 and (3.25), with , for every with , we get

(3.32)

that is, 0 is a local minimum. Moreover, by (), by now, it is evident that the functional is anticoercive in . Hence, by the regularity of , there exists which is a global maximum for the functional. Therefore, since it is not restrictive to suppose that (otherwise, there are infinitely many critical points), our conclusion follows: if , from Corollary 2.11 of [22] which ensures a third critical point different from 0 and and by standards arguments if .