  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.
