  We look for a solutions to a nonlinear, fractional Schr\"odinger equation
$$(-\Delta)^{\alpha / 2}u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on
}\mathbb{R}^N,$$ where potential $V$ is coercive or $V=V_{per} + V_{loc}$ is a
sum of periodic in $x$ potential $V_{per}$ and localized potential $V_{loc}$,
$\Gamma\in L^{\infty}(\mathbb{R}^N)$ is periodic in $x$, $\Gamma(x)\geq 0$ for
a.e. $x\in\mathbb{R}^N$ and $2<q<2^*_\alpha$. If $f$ has the subcritical
growth, but higher than $\Gamma(x)|u|^{q-2}u$, then we find a ground state
solution being a minimizer on the Nehari manifold. Moreover we show that if $f$
is odd in $u$ and $V$ is periodic, this equation admits infinitely many
solutions, which are pairwise geometrically distinct. Finally, we obtain the
existence result in the case of coercive potential $V$.
