Multiplicity results for a fractional Schrodinger equation with potentials

Abstract

We study a class of nonlinear fractional Schrodinger equations: \[ (-\Delta )^{s}u+V(x)u=f(x,u) \text { in } \mathbb{R} ^{N}, \] where $s\in (0,1)$, $N>2s$, $(-\Delta )^{s}$ stands for the fractional Laplacian. By using a variational approach, we establish the existence of at least one nontrivial solution for the above equation with a general potential $V(x)$ which is allowed to be sign-changing and a sublinear nonlinearity $f(x,u)$. Moreover, by using variational methods and the Moser iteration technique, we prove the existence of infinitely many solutions with $V(x)$ is a nonnegative potential and the nonlinearity $f(x,u)$ is locally sublinear with respect to $u$.