Multiplicity and Concentration of Solutions for Fractional Schrödinger Equations

Abstract

In this paper, we study the following fractional Schrödinger equations\[ (-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u) + \mu \xi(x) |u|^{p-2}u, \quad x \in \mathbb{R}^{N},\]where $\lambda \gt 0$ is a parameter, $V \in C(\mathbb{R}^{N})$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of two different nontrivial solutions. Moreover, the concentration of these solutions is also explored on the set $V^{-1}(0)$ as $\lambda \to \infty$. As an application, we also give the similar results and concentration phenomenons for the above problem with concave and convex nonlinearities.