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

Let $\mathrm{\Omega }$ be a bounded, smooth connected open domain in ${\mathbb{R}}^n$ with $n\ge 3$. We investigate in this paper compactness properties for the set of sign-changing solutions $v\in H_0^1(\mathrm{\Omega })$ of \begin{equation} \tag{} -Δv+h v =\left|v\right|^{2^-2}v \hbox{ in } Ω, \quad v = 0 \hbox{ on } \partial Ω \end{equation} where $h\in C^1(\bar{\mathrm{\Omega }})$ and $2^{\ast }:=2n\mathrm{/}(n-2)$. Our main result establishes that the set of sign-changing solutions of $(\ast )$ at the lowest sign-changing energy level is unconditionally compact in $C^2(\bar{\mathrm{\Omega }})$ when $3\le n\le 5$, and is compact in $C^2(\bar{\mathrm{\Omega }})$ when $n\ge 7$ provided $h$ never vanishes in $\bar{\mathrm{\Omega }}$. In dimensions $n\ge 7$ our results apply when $h>0$ in $\bar{\mathrm{\Omega }}$ 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 $(\ast )$ that holds up to the boundary. We also prove more general compactness results under perturbations of $h$.