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

Abstract

Let $\Omega$ 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 $v\in H_0^1(\Omega )$ 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 $h\in C^1(\overline{\Omega })$ and $2^{\ast }:=2n/(n-2)$. Our main result establishes that the set of sign-changing solutions of $(\ast )$ at the lowest sign-changing energy level is unconditionally compact in $C^2(\overline{\Omega })$ 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 $(\ast )$ that holds up to the boundary. We also prove more general compactness results under perturbations of $h$.

arXiv comment41 pages, comments welcome

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