Study of some nonlinear elliptic systems with variable exponents

Abstract

We study some nonlinear partial differential systems of the elliptical type involving nonhomogeneous operators. One of the major difficulties which appear during the study of these systems is the loss of compactness in the Sobolev inclusions because the system is defined on the entire space $ \mathbb {R}^N $ or it contains critical exponents. A second difficulty concerns the choice of nonlinearities, which must satisfy growth conditions. These conditions guarantee the differentiability, coercivity and semicontinuity of the energy functional associated with the system. Finally the last difficulty is linked to the construction of functional spaces with weight to ensure the structure of reflexive Banach spaces as well as the different Sobolev inclusions.