Compactness results for Sign-Changing Solutions of critical nonlinear elliptic equations of low energy

TL;DR

This paper establishes compactness properties for sign-changing solutions of a critical nonlinear elliptic equation in low energy levels, with results depending on the dimension and the behavior of the coefficient function.

Contribution

It provides new compactness results for sign-changing solutions at the lowest energy level, including a global pointwise blow-up description and extensions under perturbations.

Findings

Compactness in $C^2$ for dimensions 3 to 5.

Compactness in $C^2$ for dimensions ≥7 when $h$ is positive.

A new global pointwise description of blow-up sequences.

Abstract

Let $\Omega$ be a bounded, smooth connected open domain in $\mathbb{R}^n$ with $n\geq 3$. We investigate in this paper compactness properties for the set of sign-changing solutions $v \in H^1_0(\Omega)$ of \begin{equation} \tag{*} -\Delta v+h v =\left|v\right|^{2^*-2}v \hbox{ in } \Omega, \quad v = 0 \hbox{ on } \partial \Omega \end{equation} where $h\in C^1(\overline{\Omega})$ and $2^*:=2n/(n-2)$. Our main result establishes that the set of sign-changing solutions of $(*)$ at the lowest sign-changing energy level is unconditionally compact in $C^2(\overline{\Omega})$ when $3 \le n \le 5$, and is compact in $C^2(\overline{\Omega})$ when $n \ge 7$ provided $h$ never vanishes in $\overline{\Omega}$. In dimensions $n \ge 7$ our results apply when $h >0$ in $\overline{\Omega}$ and thus complement the compactness result of Devillanova-Solimini, Adv. Diff. Eqs. 7 (2002). Our proof is based on a new, global pointwise description of blowing-up sequences of solutions of $(*)$ that holds up to the boundary. We also prove more general compactness results under perturbations of $h$.