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

Abstract

Let $\Omega$ be a bounded, smooth domain of ${\mathbb{R}}^n,n\ge 3$ and $\lambda \ge 0$. 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 $2^{\ast }=\frac{2n}{n-2}$. When $n=3$ we investigate the existence of \emph{least-energy solutions} for this problem, that we define as having the lowest $L^{2^{\ast }}(\Omega )$ norm among all non-zero solutions. We prove that least-energy solutions of the Brézis-Nirenberg problem exist when $\lambda$ belongs to a left neighbourhood of any eigenvalue of $-\Delta$ 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 $\Omega$ when $n=3$ and $\lambda \ge {\Lambda }_1$. In order to do this we introduce, for any $\lambda \ge 0$, a new variational problem inspired from spectral-theoretic considerations which is as follows: for any $u\in L^{2^{\ast }}(\Omega ),u>0$ a.e., we consider the principal eigenvalue of $-\Delta -\lambda$ on the weighted space $L^2(\Omega ,u^{2^{\ast }-2}dx)$, whose value we then minimise over the set of normalised weights $\Vert u{\Vert }_{2^{\ast }}=1$. When $\lambda \ge {\Lambda }_1$ 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 $-\Delta$.