We look for ground state solutions to the following nonlinear Schr\"{ o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{ q-2}u\hbox{ on }\mathbb{ R}^N,$$ 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 $V_{ loc}$, $\Gamma\in L^{ \infty}(\mathbb{ R}^N)$ is periodic and $\Gamma(x)\geq 0$ for a.e. $x\in\mathbb{ R}^N$ and $2\leq q<2^*$. 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.
Categories: math.AP