Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities

Abstract

We look for ground state solutions to the following nonlinear Schrödinger equation $$-Δu + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic potential $V_{per}$ and a localized potential $V_{loc}$, $Γ\in L^{\infty}(\mathbb{R}^N)$ is periodic and $Γ(x)\geq 0$ for a.e. $x\in\mathbb{R}^N$ and $2\leq q<2^*$. We assume that $\infσ(-Δ+V)>0$, where $σ(-Δ+V)$ stands for the spectrum of $-Δ+V$ and $f$ has the subcritical growth but higher than $Γ(x)|u|^{q-2}u$, however the nonlinearity $f(x,u)-Γ(x)|u|^{q-2}u$ may change sign. Although a Nehari-type monotonicity condition for the nonlinearity is not satisfied we investigate the existence of ground state solutions being minimizers on the Nehari manifold.