Bubble solution for the critical Hartree equation in pierced domain

TL;DR

This paper proves the existence of bubble solutions to a critical Hartree equation in a pierced domain, which blow up at the origin as the puncture shrinks, extending understanding of nonlinear PDEs with singularities.

Contribution

It constructs bubble solutions for the critical Hartree equation in a domain with a small hole, revealing blow-up behavior near the puncture, a novel result in this context.

Findings

Existence of bubble solutions in pierced domains.

Solutions blow up at the origin as the hole shrinks.

Extension of critical Hartree equation analysis to punctured domains.

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.