Ground state solution of fractional Schr\"odinger equations with a general nonlinearity

Abstract

In this paper, we study the following fractional Schr\"odinger equation: \[ \left\{\begin{gathered} {(- \Delta)^s}u + mu = f(u){\text{in}}{\mathbb{R}^N}, \hfill u \in {H^s}({\mathbb{R}^N}),{\text{}}u > 0{\text{on}}{\mathbb{R}^N}, \hfill \\ \end{gathered} \right. \] where $m>0$, $N>2s$, ${(- \Delta)^s}$, $s \in (0,1)$ is the fractional Laplacian. Using minimax arguments, we obtain a positive ground state solution under general conditions on $f$ which we believe to be almost optimal.