Existence Of Solutions For Fractional P-Laplacian Equations

In this thesis,the existence and multiplicity of solutions for a class of fractional quasilinear elliptic equations with critical Choquard terms are investigated.The solutions of fractional p-Laplacian equation are obtained by finding the critical points of the corresponding energy functional with the critical point theorems.The whole thesis is divided into two parts.In the first part,the bifurcation results for the fractional p-Laplacian problem with critical Choquard terms are obtained :(?)is the critical N-spexponent in the sense of Hardy-Littlewood-Sobolev inequality,W is a bounded domain in? Nwith Lipschitz boundary.Since the nonlinear operator(-?)s pdoes not have linear eigenspaces,the standard sequence of variational eigenvalues of(-?)s pbased on the genus does not provide enough information about the structure of the sublevel sets to implement the linking construction,we use a different sequence of eigenvalues based on the? 2-cohomological index and obtain the bifurcation results for the problem by the abstract critical point theorem based on a pseudo-index.In the second part,nonlocal problems with critical Choquard nonlinearities are studied:In the second part,nonlocal problems with critical Choquard nonlinearities are studied:(?)is the fractional critical Sobolev exponent.Under some N-spsuitable assumptions,we obtain the positive solutions by the Mountain pass theorem for the subcritical and critical cases and the sign-changing solutions by the Nehari manifold for the subcritical case.