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Abstract and Applied Analysis
Volume 2012, Article ID 137379, 16 pages
http://dx.doi.org/10.1155/2012/137379
Research Article

Existence of Solutions of a Nonlocal Elliptic System via Galerkin Method

1Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2Unidade Acadêmica de Matemática e Estatística, Centro de Ciências e Tecnologia, Universidade Federal de Campina Grande, 58.109-979 Campina Grande, Brazil

Received 2 February 2012; Accepted 28 February 2012

Academic Editor: Pavel Drábek

Copyright © 2012 Alberto Cabada and Francisco Julio S. A. Corrêa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By means of the Galerkin method and by using a suitable version of the Brouwer fixed-point theorem, we establish the existence of at least one positive solution of a nonlocal elliptic N-dimensional system coupled with Dirichlet boundary conditions.

1. Introduction

This paper is devoted to the study of the nonlocal elliptic system Here, , is a bounded smooth domain, are positive numbers, , and the nonlinearities and will be defined later.

The one-dimensional counterpart of this problem has been considered by Cabada et al. in [1]. There the authors, by using dual variational methods and Leray-Schauder degree, with , and a real parameter, and under suitable assumptions on the nonlinear functions and , show the existence of solution (not necessarily positive) depending upon the parameter .

In this case, we present a different point of view from the one used in [1]. We note that, among other things, we assume .

Motivated by its many applications and the richness of the methods employed to solve it, this kind of problems has been studied by different authors when only one equation is considered, see, among others, [29]. Indeed, there is a lot of phenomena that may be modeled by equations of the form where is a nonlocal operator which, in some applications, is written in the form

See [1013] for some surveys on these equations.

In particular, steady-state solutions deliver us to elliptic equations such as which, in several cases, have a behaviour quite different from the local one

One of the most significant differences between these two types of problems is the nonexistence, in some particular cases, of maximum principles. For instance, Allegretto and Barabanova [4] consider the one-dimensional problem

It is not difficult to verify that the explicit solution of this problem is given by the expression

So, when the values of the positive parameter are small, the solution is positive. However, if is large enough, function becomes negative near the end points and . That is, in but, for large enough, the corresponding solution is not positive.

This contrasts with the local equation for which it is very well known, see [14], that, for all , function in whenever in .

Remark 1.1. We have to point out that the lack of a general maximum principle seems to be characteristic of integrodifferential operator. Indeed, in [15], the authors consider a noncooperative system, arisen in the classical FitzHugh-Nagumo systems, which serves as a model for nerve conduction. More precisely, it is studied the system where are constants and is a given function. Taking , under Dirichlet boundary condition, problem (1.9) is equivalent to the integrodifferential problem
Consider now the problem with and in .
Let be the first eigenvalue of operator in the space , and assume that . Then, for all , problem (1.11) satisfies a maximum principle, that is, under the above assumptions, the solution of (1.11) satisfies a.e in . See [15] for the proof of this result.
After that, it is proved in [16], by using semigroup theory, that this maximum principle does not hold for all . Indeed, the approach used in [16] may be used to prove that a general maximum principle for the problem (1.6) is not valid. In view of this, the method of sub- and supersolution should be used carefully by considering a relation between the growth of the nonlinearity and the parameters of the problem.

Remark 1.2. It is worthy to remark that problem (1.1) has no variational structure even in the scalar case. So the most usual variational techniques cannot be used to study it.

To attack problem (1.1), we will use the Galerkin method through the following version of the Brouwer fixed-Point Theorem whose proof may be found in Lions [17, Lemma 4.3].

Proposition 1.3. Let be a continuous function such that for some , where is the Euclidian Scalar product and is the corresponding Euclidian norm in . Then, there exists such that .

2. A Sublinear Problem

In this section, we consider the problem

Here, is a real parameter and are positive constants whose properties will be precised later.

In order to use Proposition 1.3, we have to introduce a suitable setup. First of all, we consider an orthonormal Hilbertian basis in whose norm is the usual one

Next, let be the finite dimensional vector space

equipped with the norm induced by the one in .

Thus, if , there is a unique such that

and, as a consequence,

So, the spaces and are isomorphic and isometric by

From now on, we identify, with no additional comments, via this isometry.

In order to obtain a nontrivial solution of problem (2.1), let be a constant and consider the auxiliary problem

Theorem 2.1. Assume that is a bounded domain of and the constants . Then, for all , problem (2.1) has at least one solution in .

Proof. First of all, we consider a map , defined, for all , as where we are identifying , with ,  ,  .
Now, we have that, for all , the following equations hold:
Therefore,
Denoting as for all , using the isometry between and , and the inequalities of Hölder, Poincaré, and Sobolev, we arrive at the following estimations: and, in a similar way,
We note that the positive constant depends on , but it does not depend on the rest of the parameters involved in problem (2.7).
Using the previous estimations ()–() together with the fact that , we deduce that
Now, since , there is such that
In view of Proposition 1.3, there exists that does not depend on , and a pair , such that , and satisfies the following equalities for all
So, for all , it is satisfied that
In view of the boundedness of the approximate solutions in , we have, perhaps for subsequences, that and in .
Fixing and making in the last two equalities, we obtain, after using Sobolev immersions, that the following equalities hold for all :
Since is arbitrary, the last two identities are valid for all . Consequently, is a weak solution of the auxiliary problem (2.7).
Now, since and , we have, from (2.7), that and on . This fact, together with the Dirichlet boundary conditions, says that on . In consequence, the following equalities hold for all :
That is to say, is a weak solution of problem
In particular, it satisfies the following inequalities:
Let be the unique solution of the problem
We point out that the existence and uniqueness of the solutions of each of the above equations follow from [18, 19] because .
In the sequel, we use the following comparison result, due to Ambrosetti, Brezis, and Cerami.
Lemma 2.2. [20, Lemma 3.3]. Assume that is a continuous function such that is decreasing for . Let and satisfy Then, for all .

So we conclude that and in .

Taking limits on both members of (2.16) and (2.6) as , we deduce that and , for some such that in .

Proceeding as before, by using Sobolev embeddings and elliptic regularity, we conclude that is a classical solution of the system (2.1).

Remark 2.3. We note that, from the fact that problem (1.6) is a one-dimensional particular case of problem (2.1), when , we cannot ensure that, in general, the problem (2.1) has a positive solution in .

Remark 2.4. We should point out that we may consider a more general system than (2.1). To be more precise, we may consider a system like where and are, respectively, the and . Although the proof of the existence of solution follows similar ideas as those used in Theorem 2.1, the calculations are more complicated because we have to work with Schauder’s basis in and and these spaces do not enjoy a Hilbert space structure.
If we would like to dare a little more, we may consider a system like where are, respectively, the and and , , , , , , are continuous functions on satisfying suitable conditions. In this case, we have to work in the generalized Lebesgue-Sobolev spaces and . See, for example, [21, 22] and the references therein, for more detailed information on this subject.

3. A Singular Problem

The Galerkin method may also be used to attack a singular version of the problem (2.1). More precisely, let us consider a simple version of a singular problem as with .

We should point out that other combinations of and may be considered, including convection terms like . More precisely, we may consider problems like or

However, for the sake of simplicity and to illustrate the method, we restrict our discussion to the problem (3.1).

To approach problem (3.1), we consider a nonsingular perturbation as with , to obtain approximate solutions . In this way, we avoid the singular term.

We have the following result.

Theorem 3.1. Let be real numbers and a real parameter. Then, problem (3.1) possesses a positive solution .

Proof. Reasoning as in the proof of Theorem 2.1 we obtain, for each , a solution of problem (3.4). So, since , we obtain
In view of the maximum principle, we obtain that in and in . Consequently,
If , we obtain, in view of (3.6), that , which contradicts the fact that in . Thus, and, because , the maximum principle gives in .
The solution of problem (3.1) will be obtained by studying the limit when . Thus, we may suppose that . In view of this, we obtain from (3.6) that
Let be the unique positive solution of
From the maximum principle, we deduce that in . Since is bounded, by using interior elliptic regularity, we obtain that there is such that in , for all .
Using the Galerkin method and reasoning as in the proof of Theorem 2.1, we deduce that the approximate solutions are uniformly bounded in with respect to . In consequence, we conclude that and in in the weak sense. Hence, in view of (3.8), we obtain
Invoking again the maximum principle, we get in . Since in , we conclude that in . Reasoning as before, in , and is a classical solution of problem (3.1), which finishes the proof of this result.

4. On a Superlinear Problem

At last we will make some remarks on a superlinear problem. In order to simplify the exposition, let us consider the one equation case where and . Such a function is introduced in order to ensure that the solution is nonnegative and nontrivial. Note that is a solution of and we do not have at our disposal the results in [19, 20].

Theorem 4.1. Suppose that and . Then, for each , with sufficiently small, problem (4.1) possesses a nonnegative and nontrivial solution.

Proof. First of all, let us consider the auxiliary problem
As before, let us consider , defined, for all , as where we are using the previous identifications.
For all , the following equations hold and so
Using the classical immersions, we get for all .
We now consider the function and note that
So, we may find a , sufficiently small, such that
As consequence, we can choose , small enough, such that and, taking such that , we get and note that such a does not depend on .
Proceeding as in the proof of Theorem 2.1, we obtain, for each , an approximate solution satisfying and, for all ,
Thus, for all , we have
Since we are working in the superlinear and subcritical case and using the previous arguments, we obtain that there is a function satisfying for all .
We now use the weak maximum principle to conclude that in . From the fact that , the solution is not the trivial one. Then, is a solution of our original problem (4.1).

Remark 4.2. Note that, when we are working with a problem with variational structure the function may be discarded because we may use the Mountain Pass Theorem. In this case the critical level given by this theorem is positive and so we may conclude that the solution obtained in this way is not null. However, in our case, we can not dispose of the variational techniques. In view of this and to show that we obtain a nontrivial solution, we had to introduce the function .

Acknowledgments

This work was done while the second author was working as a Visiting Professor in the Postgraduate Program in Mathematics of the Universidade Federal do Pará, Brazil. He would like to express his profound gratitude to Professor Giovany M. Figueiredo for his warm hospitality. A. Cabada was partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, Project MTM2010-15314. F. J. S. A. Corrêa was partially supported by CNPq, Grant 301603/2007-3.

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