1. Introduction

This article is devoted to presenting the results of existence of solution for a nonlinear elliptic systems of partial differential equations, in a bounded domain of with zero Dirichlet boundary conditions. These results are obtained by using Leray-Schauders topological degree and some tools of functional analysis. The corresponding scalar case considered in ^[6] wich has shown the existence of solutions to the problem where is a self-adjoint operator with compact resolvent in maps into , such that and ( a simple eigenvalue of ). In this paper we establish the existence of weak solutions for the problem

(1.1)

Where is a bounded domain in wiht smooth boundary and are continuous functions satisfying the condition below:

(1.2)

In the case where we have the existence of the solution if l is a simple eigenvalue of see ^[9]. And in ^[10] studied the case where l is not an eigenvalue i.e denotes the spectrum of The case no resonance was treated by Lakhal and Khodja (see ^[11]).

For the rest of this article, we suppose that

(1.3)

and

(1.4)

where are real positive constants.

We assume that are continuous functions satisfying the carathéodory conditions, and verifying also the growth restriction defined below:

(1.5)

where , are real positive constants.

(1.6)

and

(1.7)

Theorem 1.1. Under the assumptions (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7), there exists a solution to the problem (1.1).

2. Preliminaries

Let us consider the space

which is a Banach space endowed with the norm

such as its dual, and let us take In the sequel, and will denote the usual normes on and respectively. Recalling that the operator A, given by

defines an inverse compact on and his spectrum is formed by the sequence such that and the first eigenvalue is positive. Throughout this paper, we denote by a simple eigenvalue of is an eigenfunction associated to normalized in Pr designates the orthogonal projection of on ( is the orthogonal of in ). We recall the following proposition proved by T.Gallouet and O.Kavian (see ^[5]).

We give now a definition of weak solution.

Definition 2.1. We say is a weak solution for the system (1.1) if for any we have

(2.1)

We write the problem in the form

where is, for the element of defined by

From (1.3), (1.4) and (1.5), it is clear that the application is continuous to in For the linear problem

(2.2)

has a unique solution We note the operator who at in associates solution of (2.2). To is compact, we deduced that the operator is compact to in

The problem (1.1) is equivalent to solving the fixed point problem So we show through topological degree, the following problem has a solution

For we put The mapping H is defined to in For let us put

Let us show now that

Let we assume that We have and

We want to estimate .

where depends only on (and is given by the Poincar inequality). So

Let and show that there depending only on for such that

By definition, is a solution of

(2.3)

Taking in (2.3) we obtain

with

We have

By Rellich theorem, we deduce that the set is relatively compact in therefore is compact.

We now show that is continous.

Proposition 2.1. The mapping is continuous to in

Proof. Let converge to in

We want to show that

Let

and

To show that

seeking to pass to the limit on the following equation:

(2.4)

We know that is bounded in , because is bounded in (this is a shown in the previous step: then )

The sequence is bounded in therefore

(2.5)

Let as therefore

and

we have

But in weak. We have

Similarly we have

Then we notice that

and

thanks to Lebesgue’s convergence theorem, we deduce that

and

Similarly we have

Finally for the last term,

By dominated convergence (from (1.5) and (2.5)) we have

and consequently

Similarly we have

Passing to the limit in (2.4), we obtain

and therefore

As in where and then the mapping is continuous.

3. A Priori Bounds for Solutions of (1.1)

Let and that is to say

(3.1)

For we put ( is a primitive of , for ). As It is not difficult to show that () and

For (1.3), we have

Taking in (3.1). By assumptions (1.2), (1.3), (1.4), (1.5) and (1.6), we have

Lemma 3.1. There exist such that for all and all

Proof. To prove this lemma we assume by contradiction, that for all there exists such that

In other words, we can find a sequence such that

(3.2)

Taking

we have

and

For (1.5), we have

Moreover, by (3.2) we have

Then

that is, is bounded in

Since and the embedding is compact, we can extract a subsequence , still denoted by which converges in Let be the limit of in We have threrfore (which give and ). We also have

(3.3)

Finally, using the Poincare inequality, there is depending only on such that

Let us put

and now we show that when , which is impossible since is reduced by constant which is strictly positive.

Show that

with domination (in ), we have by the dominated convergence theorem that when .

We first show dominance.

From (1.5) and (3.3), we have

Then

We now show the convergence a.e.

We have

Let From the hypothesis (1.6) and (1.7) it follows that

Case I

If and therefore (resp if and ), and but

therefore and

Case II

If and therefore (resp if and ), and but

therefore and

Case III

If

In summary we have

It was also shown that this is contradiction with for all We have shown that there exists suth that

Now, we give the proof of our main result.

Proof. of Theorem (1.1). We have no solution to the equation on the edge of the ball such that

By invariance of the topological degree we have

is constant.

In particular . By the homotopy invariance property, we have

We infer the existence of as

Therefore is a solution of (1.1) (and theorem(1.1) is shown).

4. Conclusion

We have discussed in this article the conditions of existence for a nonlinear elliptic system. This problem has been treated by Leray-Schauder degree theory, the latter is a more tool powerful, more general and often even easier to use. The problem proposed in this paper is a generalization of working of T. Gallouët (see ^{[1, 5, 6]}).

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