Compactness results for sign-changing solutions of critical nonlinear elliptic equations of low energy

Cheikh Ali, Hussein; Premoselli, Bruno

doi:10.2140/apde.2026.19.587

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Compactness results for sign-changing solutions of critical nonlinear elliptic equations of low energy

Hussein Cheikh Ali and Bruno Premoselli

Vol. 19 (2026), No. 3, 587–626

DOI: 10.2140/apde.2026.19.587

Abstract

Let $Ω$ be a bounded, smooth connected open domain in $ℝ^{n}$ with $n \geq 3$ . We investigate compactness properties for the set of sign-changing solutions $v \in H_{0}^{1} (Ω)$ of

{\begin{matrix} - Δ v + h v = | v |^{2^{*} - 2} v & in Ω, \\ v = 0 & on \partial Ω, \end{matrix}

where $h \in C^{1} (\bar{Ω})$ and $2^{*} : = 2 n ∕ (n - 2)$ . Our main result establishes that the set of sign-changing solutions of the above system at the lowest sign-changing energy level is unconditionally compact in $C^{2} (\bar{Ω})$ when $3 \leq n \leq 5$ , and is compact in $C^{2} (\bar{Ω})$ when $n \geq 7$ provided $h$ never vanishes in $\bar{Ω}$ . In dimensions $n \geq 7$ our results apply when $h > 0$ in $\bar{Ω}$ and thus complement the compactness result of Devillanova and Solimini (2002). Our proof is based on a new, global pointwise description of blowing-up sequences of solutions of the above system that holds up to the boundary. We also prove more general compactness results under perturbations of $h$ .

Keywords

sign-changing solutions, critical nonlinear elliptic PDEs of second order, blow-up theory

Mathematical Subject Classification

Primary: 35B40, 35B44, 35B45, 35J60

Milestones

Received: 5 January 2025

Accepted: 28 April 2025

Published: 11 March 2026

Authors

Hussein Cheikh Ali
Laboratoire Paul Painlevé Université de Lille Cité Scientifique Villeneuve d’ASCQ France

Bruno Premoselli
Département de Mathématiques Université Libre de Bruxelles Bruxelles Belgium

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Vol. 19, No. 3, 2026