Bubble solution for the critical Hartree equation in pierced domain

Abstract

In this article, we establish the existence of solutions to the following critical Hartree equation \begin{align*} \begin{cases} -\Delta u=\left(\int_{\Omega_\varepsilon}\frac{u^{2_{\mu}^*}}{|x-y|^{\mu}}dy\right)u^{2_{\mu}^*-1}, &\text{ in } \Omega_\varepsilon, \\ u=0, &\text{ on } \partial\Omega_\varepsilon, \end{cases} \end{align*} where $2_{\mu}^*=\frac{2N-\mu}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $N\geq 5$, $0<\mu<4$ with $\mu$ sufficiently close to $0$, $\Omega_\varepsilon:=\Omega\backslash B(0,\varepsilon)$ and $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, which contains the origin, and $\varepsilon$ is a positive parameter. As $\varepsilon$ goes to zero, we construct bubble solution which blows up at the origin.