Nonnegative Solutions of Nonlinear Fractional Laplacian Equations

Abstract

The study of reaction-diffusion equations involving nonlocal diffusion operators has recently flourished. The fractional Laplacian is an example of a nonlocal diffusion operator which allows long-range interactions in space, and it is therefore important from the application point of view. The fractional Laplacian operator plays a similar role in the study of nonlocal diffusion operators as the Laplacian operator does in the local case. Therefore, the goal of this dissertation is a systematic treatment of steady state reaction-diffusion problems involving the fractional Laplacian as the diffusion operator on a bounded domain and to investigate existence (and nonexistence) results with respect to a bifurcation parameter. In particular, we establish existence results for positive solutions depending on the behavior of a nonlinear reaction term near the origin and at infinity. We use topological degree theory as well as the method of sub- and supersolutions to prove our existence results. In addition, using a moving plane argument, we establish that, for a class of steady state reaction-diffusion problems involving the fractional Laplacian, any nonnegative nontrivial solution in a ball must be positive, and hence radially symmetric and radially decreasing. Finally, we provide numerical bifurcation diagrams and the profiles of numerical positive solutions, corresponding to theoretical results, using finite element methods in one and two dimensions