A positive solution of asymptotically periodic Schrödinger equations with local superlinear nonlinearities

Abstract

In this paper, we investigate the following Schrödinger equation \begin{equation*} -\Delta u+V(x)u=\lambda f(u) \quad {\rm in} \ \mathbb{R}^N, \end{equation*} where $N\geq 3$, $\lambda>0$, $V$ is an asymptotically periodic potential and the nonlinearity term $f(u)$ is only locally defined for $|u|$ small and satisfies some mild conditions. By using Nehari manifold and Moser iteration, we obtain the existence of positive solutions for the equation with sufficiently large $\lambda$.

Keywords

• schrödinger equation
• positive solution
• locally defined nonlinearity
• asymptotically periodic potential