Non-local equations: existence and multiplicity results

Abstract

The main theme of my thesis is based on non-local type elliptic equations. In particular, existence of infinitely many nontrivial solutions for a class of equations driven by non-local integro-differential operator $\mathcal{L}_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions in smooth bounded domain in $\mathbb{R}^N$ is shown. Moreover, when $\mathcal{L}_K$ reduces to the fractional Laplace operator $(-\Delta)^s$, and the nonlinearity is of critical-concave type, existence of at least one sign changing solution has been established. These are then further generalized to the case of non-local equations with p-fractional Laplace operator. Existence of infinitely many nontrivial solutions for the class of equations with (p,q) fractional Laplace operator and concave-critical nonlinearities have also been studied together with existence of multiple nonnegative solutions when nonlinearity is of convex-critical type.

Also, in a different project I have studied the existence/nonexistence/qualitative properties of the positive solutions of non-local semilinear elliptic equations with critical and supercritical type nonlinearities. These are all joint published works with my supervisor Dr. Mousomi Bhakta in series of four papers.