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 $$-\Delta u+V(x)u=f(x,u)-\Gamma (x)|u{|}^{q-2}uon{\mathbb{R}}^N,$$ 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 $V_{loc}$, $\Gamma \in L^{\infty }({\mathbb{R}}^N)$ is periodic and $\Gamma (x)\ge 0$ for a.e. $x\in {\mathbb{R}}^N$ and $2\le q<2^{\ast }$. We assume that $\inf \sigma (-\Delta +V)>0$, where $\sigma (-\Delta +V)$ stands for the spectrum of $-\Delta +V$ and $f$ has the subcritical growth but higher than $\Gamma (x)|u{|}^{q-2}u$, however the nonlinearity $f(x,u)-\Gamma (x)|u{|}^{q-2}u$ 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.

References in corpus (1)

• Ground states of elliptic problems involving non homogeneous operators

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• Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearities
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• Non-local to local transition for ground states of fractional Schrödinger equations on ${\mathbb{R}}^N$