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

TL;DR

This paper proves that for a slightly supercritical Choquard equation, no single-bubble positive solutions exist as the perturbation parameter approaches zero, contrasting with the subcritical case.

Contribution

It establishes the nonexistence of single-bubble solutions in the slightly supercritical regime, extending understanding of solution behavior in nonlinear Choquard equations.

Findings

No single-bubble solutions as epsilon approaches zero

Contrasts with subcritical case where solutions exist

Provides insight into solution structure for supercritical equations

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 $\varepsilon\to 0^{+}$.