Least-energy solutions of the Brézis-Nirenberg problem in the non-coercive case in dimension $3$

Abstract

Let $Ω$ be a bounded, smooth domain of $\mathbb{R}^n, n \ge 3$ and $λ\ge 0$. We consider the celebrated Brézis-Nirenberg problem: \begin{equation}\label{eq:critlambda:abs} \tag{*} \left\{\begin{aligned} -Δu -λu & =\left|u\right|^{2^*-2}u &\hbox{ in } Ω, u & = 0 \quad \text{ in } \partial Ω, \end{aligned}\right. \end{equation} where $2^* = \frac{2n}{n-2}$. When $n=3$ we investigate the existence of \emph{least-energy solutions} for this problem, that we define as having the lowest $L^{2^*}(Ω)$ norm among all non-zero solutions. We prove that least-energy solutions of the Brézis-Nirenberg problem exist when $λ$ belongs to a left neighbourhood of any eigenvalue of $-Δ$ that we explicitly characterise by a positive mass assumption. We obtain in particular the first \emph{existence} result for the Brézis-Nirenberg problem on a general smooth bounded domain $Ω$ when $n=3$ and $λ\ge Λ_1$. In order to do this we introduce, for any $λ\ge 0$, a new variational problem inspired from spectral-theoretic considerations which is as follows: for any $u \in L^{2^*}(Ω), u>0$ a.e., we consider the principal eigenvalue of $- Δ-λ$ on the weighted space $L^2(Ω, u^{2^*-2} dx)$, whose value we then minimise over the set of normalised weights $\Vert u \Vert_{2^*} = 1$. When $λ\ge Λ_1$ this defines a new, non-smooth variational problem for which we develop a variational theory. We prove that its minimisers exist under the aforementioned positive mass assumption and that they yield \emph{least-energy} solutions. We also obtain new results in the higher-dimensional case $n \ge 4$, where we show that the energy function of the Brézis-Nirenberg problem is discontinuous exactly at the eigenvalues of $- Δ$.