Nonlinear elliptic problems in relation with the fractional Laplacian : Existence and regularity

Abstract

In this thesis, we investigate nonlinear elliptic problems involving the fractional Laplacian operator. First of all, we give a global fractional Calderón Zygmund regularity theory for the fractional Poisson problem. Our proofs are based on a pointwise estimate of the fractional gradient of the Green’s function associated to the fractional Laplacian. As an application, using a fixed point argument, we derive existence results for some fractional KPZ-type problems (equation and system), with nonlocal gradient terms. Moreover, non-existence results are also given for the problems in question. Finally, we study the impact of a Hardy potential’s presence on our previous regularity and existence results.