Fractional Schrödinger Equation in Bounded Domains and Applications

Abstract

We establish the existence and uniqueness of a positive solution to the Schrodinger equation involving the fractional Laplacian \(\Delta ^{\frac{\alpha }{2}}u=\mu \,u\) in smooth bounded domains of \(\mathbb {R}^d\) for a large class of nonnegative perturbations \(\mu \). We then use this result to give some new facts about the fractional semilinear equation \(\Delta ^{\frac{\alpha }{2}}u= u^\gamma \), \(\gamma >0\).