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

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} -Δu+au=\displaystyle\left(\int_Ω\frac{u^{6-α}(y)}{|x-y|^α}dy\right)u^{5-α}&\mbox{in}\ Ω, u>0 \ \ &\mbox{in}\ Ω, u=0 \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^{3}$ is a bounded domain with a $C^{2}$ boundary, $α\in (0,3)$, $a \in C(\overlineΩ) \cap C^{1}(Ω)$, and the operator $-Δ+ a$ 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 $V \in L^{\infty}(Ω)$, we prove that least energy solutions exist. Furthermore, we establish a refined energy estimate and describe their asymptotic profile.