Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

Abstract

In present paper, we study the fractional Choquard equation $${\epsilon }^{2s}(-\Delta {)}^{s}u+V(x)u={\epsilon }^{\mu -N}(\frac{1}{|x{|}^{\mu }}\ast F(u))f(u)+|u{|}^{2_s^{\ast }-2}u$$ where $\epsilon >0$ is a parameter, $s\in (0,1),$ $N>2s,$ $2_s^{\ast }=\frac{2N}{N-2s}$ and $0<\mu <\min \{2s,N-2s\}$. Under suitable assumption on $V$ and $f$, we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their's concentration behavior.

References in corpus (13)

• Fractional Quantum Mechanics and Levy Path Integrals
• Fractional Schrodinger equation
• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• Nonlocal diffusion and applications
• Existence of groundstates for a class of nonlinear Choquard equations
• Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R}}^N$
• Semi-classical states for the Choquard equation
• Limiting Bourgain-Brezis estimates for systems: theme and variations
• Fractional Choquard Equation with Critical Nonlinearities
• On fractional Schrodinger systems of Choquard type
• Ground state solutions for non-autonomous fractional Choquard equations
• Multiplicity and concentration results for a fractional Choquard equation via penalization method
• Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well