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

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.