The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearities

Abstract

We look for solutions to a fractional Schrödinger equation of the following form $$(-\Delta {)}^{\alpha /2}u+\left(V(x)-\frac{\mu }{|x{|}^{\alpha }}\right)u=f(x,u)-K(x)|u{|}^{q-2}uon{\mathbb{R}}^N\setminus \{0\},$$ where $V$ is bounded and close-to-periodic potential and $-\frac{\mu }{|x{|}^{\alpha }}$ is a Hardy-type potential. We assume that $V$ is positive and $f$ has the subcritical growth but not higher than $|u{|}^{q-2}u$. If $\mu$ is positive and small enough we find a ground state solution, i.e. a critical point of the energy being minimizer on the Nehari manifold. If $\mu$ is negative we show that there is no ground state solutions. We are also interested in an asymptotic behaviour of solutions as $\mu \to 0^{+}$ and $K\to 0$.

References in corpus (18)

• Fractional Quantum Mechanics and Levy Path Integrals
• Fractional Schrodinger equation
• Ten equivalent definitions of the fractional Laplace operator
• Fractional Schrödinger equation in optics
• Uniqueness and Nondegeneracy of Ground States for $(-\Delta {)}^{s}Q+Q-Q^{\alpha +1}=0$ in $\mathbb{R}$
• Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R}}^N$
• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• Some properties of the one-dimensional Lévy crystal
• Diffraction-free beams in fractional Schrödinger equation
• Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain
• Tunneling in Fractional Quantum Mechanics
• The relativistic massless harmonic oscillator
• A framework for fractional Hardy inequalities
• Ground states of a system of nonlinear Schrödinger equations with periodic potentials
• Non-linear ground state representations and sharp Hardy inequalities
• Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities
• Solutions of the fractional Schrödinger equation with sign-changing nonlinearity
• Non-homogeneous fractional Schrödinger equation

Cited by in corpus (3)

• Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree-Fock theory
• Bound state solutions for non-autonomous fractional Schrödinger-Poisson equations with critical exponent
• Schrödinger-type equations with sign-changing nonlinearities: a survey