Multiplicity results for fractional elliptic equations and system of equations

Abstract

The major theme of this thesis is the study of multiplicity results for fractional elliptic equations and the system of equations. The thesis is mainly divided into three parts. In the first part, the existence and multiplicity of positive solutions for perturbed nonlocal scalar field equation with subcritical nonlinearity and nonhomogeneous terms have been studied, and the global compactness result has been proved. The second part deals with Fractional Hardy-Sobolev equation involving critical nonlinearity and nonhomogeneous term. The existence of at least two positive solutions is obtained provided the corresponding nonhomogeneous terms are small enough in the dual space norm. Besides the profile decomposition for the Palais-Smale sequences of the associated energy functional has been accomplished. The third part comprises of the study of nonhomogeneous weakly coupled nonlocal system of equations with critical and subcritical nonlinearities. Firstly, the existence of a positive solution exploiting the local geometry of the associated functional near the origin is achieved. Then proving the global compactness result (which gives the complete description of the associated Palais Smale sequences for the system), the existence of two positive solutions is obtained under some suitable conditions on the nonhomogeneous terms. In addition, considering the corresponding homogeneous system, uniqueness for the ground state solution has been proved.