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

Abstract

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 $V$.