Singularly perturbed critical Choquard equations

TL;DR

This paper investigates the semiclassical limit of singularly perturbed critical Choquard equations, establishing existence, multiplicity, and concentration behavior of solutions using variational methods.

Contribution

It introduces new results on existence and multiplicity of solutions for the critical Choquard equation with singular perturbations and analyzes their concentration phenomena.

Findings

Existence of ground state solutions for the critical Choquard equation.

Multiple semi-classical solutions with concentration behavior.

Solutions characterized by variational methods.

Abstract

In this paper we study the semiclassical limit for the singularly perturbed Choquard equation $$ -\vr^2\Delta u +V(x)u =\vr^{\mu-3}\Big(\int_{\R^3} \frac{Q(y)G(u(y))}{|x-y|^\mu}dy\Big)Q(x)g(u) \quad \mbox{in $\R^3$}, $$ where $0<\mu<3$, $\vr$ is a positive parameter, $V,Q$ are two continuous real function on $\R^3$ 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 $\R^3$. Next we establish existence and multiplicity of semi-classical solutions and characterize the concentration behavior by variational methods.