An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part

Abstract

We consider the semilinear Schrödinger equation

$$\{\begin{array}{c}-△u+V\left(x\right)u=f(x,u)\text{,}x\in {\mathbb{R}}^N\text{,}\hfill \\ u\in H^1\left({\mathbb{R}}^N\right)\text{,}\hfill \end{array}$$

where $V\left(x\right)$ is asymptotically periodic and sign-changing, $f(x,u)$ is asymptotically periodic in $x$, and asymptotically linear as $\left|u\right|\to \infty$. Under some weak assumptions on $V$ and $f$, we prove that the above problem has a nontrivial solution.