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} -Δu=\displaystyle\Big(\int\limits_Ω\frac{u^{2^*_α+\varepsilon}(y)}{|x-y|^α}dy\Big)u^{2^*_α-1+\varepsilon},\quad u>0\ \ &\mbox{in}\ Ω, \quad \ \ u=0 \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $N\ge 3$, $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $\alpha \in (0,N)$, $2_{\alpha }^{\ast }:=\frac{2N-\alpha }{N-2}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and $\epsilon >0$ 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 $\epsilon \to 0^{+}$.

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