Study of Some Nonlinear Elliptic Systems in Integer and Fractional Modular Spaces

Abstract

This thesis covers three main axes: The first one seeks to the study of local nonlinear elliptic systems, the second concerns some classes of fractional systems, while the third focus on fractional systems in generalized spaces. We start by studying the existence of solutions for a nonlinear elliptic system involving the (p(.), q(.))-Laplacian operators with weight. The approach is based on sub-supersolutions method combined with Schauder fixed point theorem. As an application, we give the results for a special weight functions w1(x)=|x|ap and w2(x)=|x|aq(.). Next, using the three critical points theorem, we deal with study of a class of Kirchhoff type systems driven by fractional order operators. At first, we establish the existence of three solutions for a fractional elliptic system of p Kirchhoff type. Also, we generalize the result to the fractional Sobolev spaces with variable exponents. In the same framework of fractional Sobolev spaces with variable exponents, we investigate the existence and multiplicity of solutions for a nonlinear elliptic system of (p(x,.), q(x,.))-Laplacian operators with nonstandard growth conditions. Our main tool is based on Mountain Pass theorem. Finally, we set four results of existence in which we generalize some of the above results to RN. Our main tools come from a direct variational methods, the Mountain Pass theorem, the symmetric Mountain Pass theorem and the Fountain theorem in critical point theory. The obtained results significantly contribute the study of Kirchhoff type systems in the sense that our situation covers not only differential operators of fractional order but also nonhomogeneous differential operators in Sobolev spaces with variable exponents.