Existence and uniqueness of solution for a nonlinear fractional problem involving the distributional Riesz derivative

In this article, we prove the existence of weak solutions for the following nonlinear problem that contains a nonlocal operator -D-s center dot (AD(s)(.)) which is defined by the distributional Riesz fractional derivatives and a measurable, bounded, positive definite matrix A(x). {-D-s center dot(A(x)D(s)u(x))=f(x,u(x)) in Omega, u=0 on R-n\Omega. where Omega subset of R-n is a bounded open set with a Lipschitz boundary, s is an element of (0, 1) and n > 2s. Under some suitable conditions on the nonlinear term f and the matrix A(x), it has shown that this problem has at least one weak solution u. We use in our proof the Leray-Schauder degree method to prove the existence of weak solutions and the Banach fixed point theorem to prove the uniqueness of weak solution in a particular case.