Solutions of the fractional Schr\"odinger equation with sign-changing nonlinearity

We look for a solutions to a nonlinear, fractional Schr"odinger equation $$(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}u+V(x)u=f(x,u)-\mathrm{\Gamma }(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u\text{ on }{\mathbb{R}}^N,$$ where potential $V$ is coercive or $V=V_{per}+V_{loc}$ is a sum of periodic in $x$ potential $V_{per}$ and localized potential $V_{loc}$, $\mathrm{\Gamma }\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ is periodic in $x$, $\mathrm{\Gamma }(x)\ge 0$ for a.e. $x\in {\mathbb{R}}^N$ and $2<q<2_{\alpha }^{\ast }$. If $f$ has the subcritical growth, but higher than $\mathrm{\Gamma }(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u$, 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 $V$.