Singularly perturbed critical Choquard equations

Abstract

In this paper we study the semiclassical limit for the singularly perturbed Choquard equation $$ -\vr^2Δu +V(x)u =\vr^{μ-3}\Big(\int_{\R^3} \frac{Q(y)G(u(y))}{|x-y|^μ}dy\Big)Q(x)g(u) \quad \mbox{in $\R^3$}, $$ where $0<\mu <3$, $\vr$ is a positive parameter, $V,Q$ are two continuous real function on ${\mathbb{R}}^3$ and $G$ is the primitive of $g$ which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on the nonlinearity $g$, we first establish the existence of ground states for the critical Choquard equation with constant coefficients in ${\mathbb{R}}^3$. Next we establish existence and multiplicity of semi-classical solutions and characterize the concentration behavior by variational methods.

References in corpus (3)

• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation
• Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in ${\mathbb{R}}^2$

Cited by in corpus (1)

• On the critical Choquard equation with potential well