Nontrivial Solutions for Schrödinger Equation with Local Super-Quadratic Conditions

Abstract

This paper is dedicated to studying the semilinear Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \quad x\in {\mathbb {R}}^{N},\\ u\in H^{1}({\mathbb {R}}^{N}), \end{array}\right. \end{aligned}$$

where $V\in \mathcal {C}(\mathbb {R}^N, \mathbb {R})$ is sign-changing and either periodic or coercive and $f\in \mathcal {C}(\mathbb {R}^N\times \mathbb {R}, \mathbb {R})$ is subcritical and local super-linear (i.e. allowed to be super-linear at some $x\in \mathbb {R}^N$ and asymptotically linear at other $x\in \mathbb {R}^N$). Instead of the common condition that $\lim _{|t|\rightarrow \infty }\frac{\int _{0}^{t} f(x, s)\mathrm {d}s}{t^2}=\infty$ uniformly in $x\in \mathbb {R}^N$, we use a local super-quadratic condition $\lim _{|t|\rightarrow \infty }\frac{\int _{0}^{t} f(x,s)\mathrm {d}s}{t^2}=\infty$ a.e. $x\in G$ for some domain $G\subset \mathbb {R}^N$ to show the existence of nontrivial solutions for the above problem.