Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well

Abstract

In this paper we study the non-linear fractional Schrodinger equation with steep potential well \begin{eqnarray} (-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u) in R^{n}, u\in H^{\alpha}(R^n), \end{eqnarray} where $(-\Delta)^\alpha$ ($\alpha \in (0,1)$) denotes the fractional Laplacian, $\lambda$ is a parameter, $V\in C(\mathbb{R}^n)$ and $V^{-1}(0)$ has nonempty interior. Under some suitable conditions, the existence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is also explored.