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We look for ground state solutions to the following nonlinear Schr\"{o}dingerequation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on}\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is thesum 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 $f$ hasthe subcritical growth but higher than $\Gamma(x)|u|^{q-2}u$, however thenonlinearity $f(x,u)-\Gamma(x)|u|^{q-2}u$ may change sign. Although aNehari-type monotonicity condition for the nonlinearity is not satisfied weinvestigate the existence of ground state solutions being minimizers on theNehari manifold.

Nonlinear Schr\"odinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities