Abstract

Using some potential theory tools and the Schauder fixed point theorem, we prove the existence of positive continuous solutions with a precise global behavior for the competitive semilinear elliptic system , in an exterior domain of , subject to some Dirichlet conditions, where , , , and the potentials are nonnegative and satisfy some hypotheses related to the Kato class .

1. Introduction

The study of nonlinear elliptic systems has a strong motivation, and important research efforts have been made recently for these systems aiming to apply the results of existence and asymptotic behavior of positive solutions in applied fields. Coupled nonlinear Shrödinger systems arise in the description of several physical phenomena such as the propagation of pulses in birefringent optical fibers and Kerr-like photorefractive media, see [1, 2]. Stationary elliptic systems arise also in other physical models like non-Newtonian fluids: pseudoplastic fluids and dilatant fluids [3, 4], non-Newtonian filtration [5], and the turbulent flow of a gas in porous medium [6, 7]. They also describe other various nonlinear phenomena such as chemical reactions, pattern formation, population evolution where, for example, and represent the concentrations of two species in the process. As a consequence, positive solutions of such are of interest.

For some recent results on the qualitative analysis and the applications of positive solutions of nonlinear elliptic systems in both bounded and unbounded domains we refer to [8–15] and the references therein.

In these works various existence results of positive bounded solutions or positive blowing-up ones (called also large solutions) have been established, and a precise global behavior is given. We note also that several methods have been used to treat these nonlinear systems such as sub- and super-solutions method, variational method, and topological methods.

In this paper, we consider an unbounded domain in with a nonempty compact boundary consisting of finitely many Jordan curves and noncontaining zero. We fix two nontrivial nonnegative continuous functions and on and some nonnegative constants, , , , such that and , and we will deal with the existence of a positive continuous solution (in the sense of distributions) to the system: where , , , and are two nonnegative functions satisfying some hypotheses related to the Kato class defined and studied in [16, 17] by means of the Green function of the Dirichlet Laplacian in .

Our method is based on some potential theory tools which we apply to give an existence result for equations by an approximation argument, then we use the result for equations to prove, by means of the Schauder fixed point theorem, the existence result for the system (1.1).

As far as we know, there are no results that contain existence of positive solutions to the elliptic system (1.1) in the case where and and the weights and are singular functions.

The study of (1.1) is motivated by the existence results obtained in [18] to the following system where are nonnegative constants, the functions are nondecreasing and continuous.

More precisely, it was shown in [18] that if the functions and belong to the Kato class , then there exist and such that for each and , the system (1.2) has a positive continuous solution having the global asymptotic behavior of the unique solution of the associated homogeneous system, where we have denoted by and the functions and are the harmonic functions defined, respectively, by (1.4) and (1.5) below.

The system of two equations in (1.1) has been treated in exterior domains of , in [19], and existence of positive bounded continuous solutions is established. The main difficulty in the present work is the case of the domain . More precisely, the function defined by (1.5) behaves as at infinity for , unlike the case where this function is bounded at infinity.

Throughout this paper, we denote by the unique bounded continuous solution of the Dirichlet problem where is a nonnegative continuous function on .

The function is defined on by First, we recall the following result about this function .

Proposition 1.1 (see [16]). The function defined by (1.5) is harmonic and positive in and satisfies

Taking into account these notations, we use some potential theory tools and an approximating sequence in order to prove the following first result concerning the existence of a unique positive continuous solution to the boundary value problem: where , is a nontrivial nonnegative continuous function on and are two nonnegative constants with . More precisely we establish the following.

Theorem 1.2. Let be a nonnegative function such that the function belongs to the Kato class . Then problem (1.7) has a unique positive continuous solution satisfying for each where is defined in (1.3) and the constant .

Next we exploit this result to prove the existence of a positive continuous solution to the system (1.1). For this aim we denote by and we need to assume the following hypothesis on the functions and .(H) and are nonnegative measurable functions in such that are in .

Using the Schauder fixed point, we prove the following main result.

Theorem 1.3. Under the hypothesis , the problem (1.1) has a positive continuous solution satisfying for each in where , are defined by (1.3) and .

In order to state these results and for the sake of completeness, we give in the sequel some notations, and we recall some properties of the Kato class studied in [16, 17].

Let us denote by the set of Borel measurable functions in and by the set of nonnegative ones. We denote also by the set of continuous functions in having limit zero at , by and by . We note that is a Banach space endowed with the uniform norm .

First we recall that if is a nonnegative continuous function on , then from [20, page 427] the function and satisfies .

For any in , we denote by the Green potential of defined on by and we recall that if and , then we have in the distributional sense (see [21, page 52]) Furthermore, we recall that for , the potential is lower semicontinuous in and if with and , then for .

Let be the Brownian motion in and be a probability measure on the Brownian continuous paths starting at . For any function , we define the kernel by where is the expectation on and .

If is a nonnegative function in such that , the kernel satisfies the following resolvent equation (see [21, 22]) So for each such that , we have and for each , we have Now we recall the definition of the Kato class which contains in particular a wider class of singular functions near the boundary of the domain .

Definition 1.4 (see [16]). A Borel measurable function in belongs to the Kato class if where and denotes the Euclidian distance from to the boundary of .

This Kato class is rich enough as it can be seen in the following example.

Example 1.5 (see [16]). Let for . Then

Remark 1.6. Let and such that . Then using the Hölder inequality and the same arguments as in the proof of the precedent example it follows that for each , the function defined in by belongs to .

Next, we recall some properties of .

Proposition 1.7 (see [16, 17]). Let be a nonnegative function in . Then one has(i). (ii)The function is in . In particular .(iii). (iv)For any nonnegative superharmonic function in and all , one has

The following compactness results will be used and they are proved, respectively, in [17] and [16].

Proposition 1.8 (see [17, Lemma 3.1]). Let be a positive harmonic function in , which is continuous and bounded in and let be a nonnegative function belonging to . Then the family of functions: is uniformly bounded and equicontinuous on . Consequently, it is relatively compact in .

Proposition 1.9 (see [16, Lemma 4.3]). Let be the function defined by (1.5) and let be a nonnegative function in . Then the family of functions: is uniformly bounded and equicontinuous on , and consequently it is relatively compact in .

As a consequence of these Propositions, we obtain the following.

Corollary 1.10. Let be a nonnegative function in . Then the family of functions: is relatively compact in .

Proof. Since then the result follows from Propositions 1.8 and 1.9.

The following result will play an important role in the proofs of Theorems 1.2 and 1.3.

Proposition 1.11 (see [17, Proposition 2.9]). Let be a nonnegative superharmonic function in and be a nonnegative function in . Then for each such that , one has

2. Proof of Theorem 1.2

First we give two Lemmas that will be used for uniqueness.

Lemma 2.1 (see [23]). Let be a function in and be a nonnegative superharmonic function in . Then for all such that and , one has .

Lemma 2.2. Let be a nonnegative continuous function in . Then

Proof. Let be a nonnegative continuous solution of (1.7). First, we will prove that in . Since is bounded, then . Consequently, for there exists such that This implies that the function satisfies Hence by [20, page 465], we get in . Since is arbitrary, this implies that for each . Now, since , then . Hence it follows from Propositions 1.8 and 1.9 that and belong to with boundary value zero, which implies that belongs to with boundary value zero. So, belongs to with boundary value zero. Consequently, using Corollary 7 page 294 in [20], we deduce that the function is a classical harmonic in with boundary value zero and satisfying . Thus by [20, page 419], we have in . So and this proves necessity.
Now, we prove sufficiency. Let be a nonnegative continuous function in satisfying the integral equation . Since is nonnegative and , then and . This implies, by using Propositions 1.8 and 1.9 that and are in with boundary value zero. Consequently, is in with boundary value zero. Hence, (in the sense of distributions) and is a solution of (1.7).

Now we prove Theorem 1.2

Proof of Theorem 1.2. First we show that problem (1.7) has at most one continuous solution. Let be two continuous solutions of (1.7). Then, by Lemma 2.2 we have and in . Put and if and whenever . Then we have and in . Using Lemma 2.1, we deduce that and so .
Next, we prove the existence of a positive continuous solution to (1.7). We recall that and . Put where the constant is defined in Proposition 1.7. We define the nonempty closed bounded convex set by Let be the operator defined on by We will prove that maps to itself. Indeed, for each , we have On the other hand, since the function , we deduce by Proposition 1.11 that . Hence, .
Next, we prove that is nondecreasing on . Let such that . Since for each , the function is nondecreasing on we deduce that Now, we consider the sequence defined by and . Clearly and . Thus, using the fact that is invariant under and the monotonicity of , we deduce that Hence, the sequence converges to a measurable function . Therefore, by applying the monotone convergence theorem, we deduce that satisfies the following equation: or equivalently Applying the operator on both sides of (2.10), we deduce by using (1.16) and (1.17) that Now, let us verify that is a solution of the problem (1.7). Since , then by Proposition 1.7, we have .
Now, using the following inequality: and the continuity of in , we obtain that . Using Proposition 1.7 and (2.12), we obtain for each which gives . Thus, by applying on both sides of (2.11), we deduce that is a solution of Using (2.12), we obtain that and are in . So and are two lower semicontinuous functions. On the other hand, by Proposition 1.8 we have and by Proposition 1.9 the function . So . Thus is also an upper semicontinuous function. Consequently, . Thus . Therefore . Now, using Propositions 1.1 and 1.9, we deduce that . In addition, since is bounded in , we deduce from Proposition 1.7  . So that . This in turn implies that . Which together with (2.11) imply that . On the other hand, we have Using Propositions 1.9, 1.8, and 1.1, we deduce that tends to zero as and consequently . This implies that is a positive continuous solution of (1.7). This completes the proof of Theorem 1.2.