Solutions of the fractional Schrödinger equation with sign-changing nonlinearity

Abstract

We look for a solutions to a nonlinear, fractional Schrödinger equation $$(-\Delta {)}^{\alpha /2}u+V(x)u=f(x,u)-\Gamma (x)|u{|}^{q-2}uon{\mathbb{R}}^N,$$ 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 $x$, $\Gamma (x)\ge 0$ for a.e. $x\in {\mathbb{R}}^N$ and $2<q<2_{\alpha }^{\ast }$. If $f$ has the subcritical growth, but higher than $\Gamma (x)|u{|}^{q-2}u$, 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 $V$.

References in corpus (5)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• Diffraction-free beams in fractional Schrödinger equation
• Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities
• Propagation dynamics of a light beam in fractional Schr\"odinger equation
• Fractional nonlinear Schrödinger equations with singular potential in ${\mathbf{R}}^n$

Cited by in corpus (3)

• The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearities
• Systems of coupled Schrödinger equations with sign-changing nonlinearities via classical Nehari manifold approach
• Non-local to local transition for ground states of fractional Schrödinger equations on ${\mathbb{R}}^N$