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} -Δu=\left(\int_{Ω_\varepsilon}\frac{u^{2_μ^*}}{|x-y|^μ}dy\right)u^{2_μ^*-1}, &\text{ in } Ω_\varepsilon, \\ u=0, &\text{ on } \partialΩ_\varepsilon, \end{cases} \end{align*} where $2_{\hat{I}}{¼}^{\ast }=\frac{2N-\hat{I}¼}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $N\ge 5$, $0<\hat{I}¼<4$ with $\hat{I}¼$ sufficiently close to $0$, $\hat{I}{\text{c◯}}_{\epsilon }:=\hat{I}\text{c◯}\backslash B(0,\epsilon )$ and $\hat{I}\text{c◯}$ is a bounded smooth domain in ${\mathbb{R}}^N$, which contains the origin, and $\epsilon$ is a positive parameter. As $\epsilon$ goes to zero, we construct bubble solution which blows up at the origin.