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SUR LES SYSTEMES ELLIPTIQUES QUASI-LINEAIRES ET ANISOTROPIQUES AVEC EXPOSANTS CRITIQUES DE SOBOLEV.

Abstract : The objective of this thesis is to study the existence, multiplicity and the behaviour of positive solutions to some systems involving the p-Laplacian or the anisotropic operators either in subcritical and critical cases.
In the 1st chapter we are interested in the following system (S):
\begin{eqnarray}
\left\{\begin{array}{lll}
\Delta_p u&=& f(x,u,v), \quad\mbox{dans}\quad\Omega\\
\Delta_q v&=& g(x,u,v), \quad\mbox{dans}\quad\Omega
\end{array}
\right.
\end{eqnarray}
with $f$ and $g$ present two subcritical terms. We construct two Palais-Smale sequences on the Nehari manifold converging strongly in $W{1,p}(\Omega)\times W{1,q}(\Omega)$ to two different positive solutions.
In the 2nd chapter, we consider the same class of system (S) with critical conditions in $\mathbb{R}^N$. In the difference with the chapter 1, we find only one nonnegative solution and for $p=q$ we succeed to find another solution.
In the chapter 3, we generalize the study of Brézis-Nirenberg to an equation and then to a system of type (S). We give a more general definition to the notion of the critical level.
The last chapter deals with a new class of systems of anisotropic elliptic equations (power-like depends on the direction) with different powers,
so that the underlying functional-analytic framework involves aniotropic Sobolev spaces. We show the existence and regularity results of weak solutions of the system then an existence of a solution provided that the system has a sub and super-solution.
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Contributor : Khalid Adriouch <>
Submitted on : Friday, October 12, 2007 - 3:57:45 AM
Last modification on : Wednesday, November 29, 2017 - 10:29:59 AM
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Khalid Adriouch. SUR LES SYSTEMES ELLIPTIQUES QUASI-LINEAIRES ET ANISOTROPIQUES AVEC EXPOSANTS CRITIQUES DE SOBOLEV.. Mathématiques [math]. Université de La Rochelle, 2007. Français. ⟨tel-00178675⟩

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