Index Theory and Multiple Solutions for a Fractional Laplacian Equation With an Asymptotically Linear Term

Abstract

In this paper, we investigate the existence and multiplicity of solutions for the asymptotically linear fractional Laplacian equation with homogeneous Dirichlet boundary conditions. We will construct an index theory for the associated linear fractional Laplacian equation. Using results from critical point theory, we show how the behavior of the nonlinearity near zero and at infinity affects the number of solutions via the index.