Blow-up asymptotics for a critical Hartree-type Br\'{e}zis--Nirenberg problem in dimension three

Abstract

In this paper, we study the following critical Hartree problem \begin{equation}\label{equationabstract} \begin{cases} \displaystyle-\Delta u+(Q+\varepsilon V)u= A_{3,\mu} \left(\int_{\Omega}\frac{u^{6-\mu}(y)}{|x-y|^{\mu}}dy\right) u^{5-\mu} &\mathrm{~in~}\Omega,\\ \displaystyle u>0&\mathrm{~in~}\Omega,\\u=0&\mathrm{~on~}\partial\Omega,\end{cases} \end{equation} where $\Omega\subset\mathbb{R}^3$ is a bounded open set, $0<\mu<2$, the exponent $6-\mu$ is the upper critical exponent in the sense of the Hardy--Littlewood--Sobolev inequality and $A_{3,\mu}>0$ is a normalization constant. The function $Q$ is assumed to be critical in the sense of Hebey and Vaugon, and the solutions $u_{\varepsilon}$ of \eqref{equationabstract} are assumed to be an optimizing sequence for the Hardy--Littlewood--Sobolev inequality. Under a natural nondegeneracy assumption, we derive a precise asymptotic expansion of \(u_{\varepsilon}\), determine the exact blow-up rate, and identify the concentration point. We also obtain the pointwise blow-up behavior both near and away from the concentration point.