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

Jinkai Gao

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 $N\geq 3$, $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$, $α\in (0,N)$, $2^*_α:=\frac{2N-α}{N-2}$ 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 $\varepsilon\to 0^{+}$.

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 , is a smooth bounded domain in , , is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and 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 .
Paper Structure (4 sections, 12 theorems, 117 equations)

This paper contains 4 sections, 12 theorems, 117 equations.

Table of Contents

  1. Introduction and Main result
  2. Preliminaries
  3. Some useful estimates
  4. Proof of Theorem \ref{['thm-1']}

Key Result

Theorem 1.1

Assume that $\alpha\in(0,\min\{4,N\})$, then problem slightly supercritical choquard equation has no solution $u_{\varepsilon}$ such that with $\xi_{\varepsilon}\in\Omega$, $\lambda_{\varepsilon}\in{\mathbb R}^{+}$, $\lambda_{\varepsilon}d(\xi_{\varepsilon},\partial\Omega)\to+\infty$ and $w_{\varepsilon}\to 0$ in $H^{1}_{0}(\Omega)$ as $\varepsilon\to0^{+}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • ...and 13 more