Study of a System of Convection-Diffusion-Reaction

Samira Lecheheb, Hakim Lakhal, Maouni Messaoud, Kamel Slimani

Study of a System of Convection-Diffusion-Reaction

Samira Lecheheb1, Hakim Lakhal1,, Maouni Messaoud1, Kamel Slimani1

1Université de Skikda, B.P.26 route d’El-Hadaiek, 21000, Algérie


In this article, we are interested in the study of the existence of weak solutions of boundary value problem for the nonlinear elliptic system , where Ω is a bounded domain in and are continuous functions . We use the Leray-Schauder degree theory under not linear for the three reasons: the terms of diffusion, convection and reaction, and the following condition on the last term f and and

Cite this article:

  • Samira Lecheheb, Hakim Lakhal, Maouni Messaoud, Kamel Slimani. Study of a System of Convection-Diffusion-Reaction. International Journal of Partial Differential Equations and Applications. Vol. 4, No. 2, 2016, pp 32-37.
  • Lecheheb, Samira, et al. "Study of a System of Convection-Diffusion-Reaction." International Journal of Partial Differential Equations and Applications 4.2 (2016): 32-37.
  • Lecheheb, S. , Lakhal, H. , Messaoud, M. , & Slimani, K. (2016). Study of a System of Convection-Diffusion-Reaction. International Journal of Partial Differential Equations and Applications, 4(2), 32-37.
  • Lecheheb, Samira, Hakim Lakhal, Maouni Messaoud, and Kamel Slimani. "Study of a System of Convection-Diffusion-Reaction." International Journal of Partial Differential Equations and Applications 4, no. 2 (2016): 32-37.

Import into BibTeX Import into EndNote Import into RefMan Import into RefWorks

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


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


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




where are real positive constants.

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


where , are real positive constants.




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


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


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


Taking in (2.3) we obtain


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



To show that

seeking to pass to the limit on the following equation:


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


Let as therefore


we have

But in weak. We have

Similarly we have

Then we notice that


thanks to Lebesgue’s convergence theorem, we deduce that


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


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



we have


For (1.5), we have

Moreover, by (3.2) we have


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


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


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


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]).


[1]  A. Fattah, T. Gallouët, H. lakehal. An Existence proof for the stationary compressible stokes problem, Ann. Fac. Sci. de Toulouse Math. 6, 4 (2014), 847-875.
In article      View Article
[2]  D. G. de Figueiredo. Semilinear elliptic systems. Lectures at the international school on nonlinear differential equations, Trieste-Italy, October (2006).
In article      
[3]  D. G. de Figueiredo, B. Ruf. Elliptic systems with nonlinearities of arbitrary growth, Mediterr, J. Math, 1 (2004), 417-431.
In article      View Article
[4]  D. G. de Figueiredo, J. Yang. A priori bounds for positive solutions of a non-variational elliptic systems, Commun. Partial. Differ. Equations. 26 (2001), 2305-2321.
In article      View Article
[5]  Th. Gallouet, O. Kavian. Résultats d’existence et de non-existence pour certains problèmes demilinéares à l’infini, Ann. Fac. Sci. de Toulouse Math. 5, 3 (1981), 201-246.
In article      View Article
[6]  Th. Gallouet, O. Kavian. Resonace for jumping non-linearities, Journal of Partial Differential Equations. 7, 3 (1982), 325-342.
In article      View Article
[7]  J. Kazdan, F.Waxusn; Remarks on some quasilinear elliptic equations, Comm. Pure. Appl. Math 28 (1975), 567-597.
In article      View Article
[8]  O. Kavian. Introduction à la théorie des points critiques et Applications aux Problémes Elliptiques, Springer Verlag, Math. Appl, Vol 13, 1993.
In article      
[9]  H. Lakehal, B. Khodja, W. Gharbi. Existence results of nontrivial solutions for a semi linear elliptic system at resonance, Journal of Advanced Research in Dynamical and Control Systems Vol. 5, Issue. 3, 2013, pp. 1-12.
In article      
[10]  A. Moussaoui, B. Khodja. Existence results for a class of semilinear elliptic systems, Journal of Partial Differential Equations, 22 (2009), 111-126.
In article      
[11]  H. Lakhal, B. Khodja. Elliptic systems at resonance for jumping non- linearities, Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 70, pp. 1-13.
In article      
  • CiteULikeCiteULike
  • MendeleyMendeley
  • StumbleUponStumbleUpon
  • Add to DeliciousDelicious
  • FacebookFacebook
  • TwitterTwitter
  • LinkedInLinkedIn