Existence of standing waves for Schrodinger equations involving the fractional Laplacian

Abstract

We study a class of fractional Schrodinger equations of the form $$ \varepsilon^{2\alpha}(-\Delta)^\alpha u+ V(x)u = f(x,u) \quad\text{in } \mathbb{R}^N, $$ where $\varepsilon$ is a positive parameter, $0 < \alpha < 1$, $2\alpha < N$, $(-\Delta)^\alpha$ is the fractional Laplacian, $V:\mathbb{R}^{N}\to \mathbb{R}$ is a potential which may be bounded or unbounded and the nonlinearity $f:\mathbb{R}^{N}\times \mathbb{R}\to \mathbb{R}$ is superlinear and behaves like $|u|^{p-2}u$ at infinity for some $2<p< 2^*_\alpha:=2N/(N-2\alpha)$. Here we use a variational approach based on the Caffarelli and Silvestre's extension developed in [3] to obtain a nontrivial solution for $\varepsilon$ sufficiently small.

Keywords

• Variational methods
• critical points
• fractional Laplacian