Nonexistence of single-bubble solutions for a slightly supercritical Choquard equation

Abstract

In this paper, we consider the existence of positive solutions to the following slightly supercritical Choquard equation \begin{equation*} \begin{cases} -\Delta u=\displaystyle\Big(\int\limits_{\Omega}\frac{u^{2^*_{\alpha}+\varepsilon}(y)}{|x-y|^\alpha}dy\Big)u^{2^*_{\alpha}-1+\varepsilon},\quad u>0\ \ &\mbox{in}\ \Omega, \quad \ \ u=0 \ \ &\mbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $N\geq 3$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $\alpha\in (0,N)$, $2^*_{\alpha}:=\frac{2N-\alpha}{N-2}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and $\varepsilon>0$ is a small parameter. In contrast with the slightly subcritical Choquard equation studied by Chen and Wang (Calculus of Variations and Partial Differential Equations, 63:235, 2024), we find that there is no chance to construct a family of single-bubble solutions as $\varepsilon\to 0^{+}$.