Existence and asymptotics for the upper critical Choquard equation in dimension three

TL;DR

This paper investigates the existence and asymptotic behavior of least energy solutions to the upper critical Choquard equation in three dimensions, establishing equivalences related to the Robin function and analyzing solution profiles under perturbations.

Contribution

It introduces a new critical function for the problem, proves solution existence under perturbations, and provides refined energy estimates and asymptotic profiles.

Findings

Existence of least energy solutions is equivalent to a positivity condition of the Robin function.

Solutions exist under perturbations of the critical operator.

Refined energy estimates and asymptotic profiles of solutions are established.

Abstract

In this paper, we are interested in the existence and asymptotic behavior of least energy solutions to the upper critical Choquard equation \begin{equation*} \begin{cases} -\Delta u+au=\displaystyle\left(\int_{\Omega}\frac{u^{6-\alpha}(y)}{|x-y|^\alpha}dy\right)u^{5-\alpha}&\mbox{in}\ \Omega, u>0 \ \ &\mbox{in}\ \Omega, u=0 \ \ &\mbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^{3}$ is a bounded domain with a $C^{2}$ boundary, $\alpha \in (0,3)$, $a \in C(\overline{\Omega}) \cap C^{1}(\Omega)$, and the operator $-\Delta + a$ is coercive. We first establish that the following three properties are equivalent: the existence of least energy solutions, the validity of a strict inequality in the associated minimization problem, and the positivity of the Robin function somewhere in the domain. This leads naturally to the definition of a critical function $a$. Under the perturbation $a \mapsto a + \varepsilon V$ with $a$ critical and $V \in L^{\infty}(\Omega)$, we prove that least energy solutions exist. Furthermore, we establish a refined energy estimate and describe their asymptotic profile.