Spectral analysis of multi-bubble solutions to the Brezis-Nirenberg problem

We investigate weighted spectral limits for separated multi-bubble configurations and refined spectral asymptotics for the Brezis–Nirenberg problem $$-\mathrm{\Delta }u=u^p+\epsilon u^q,u>0,\text{in }\mathrm{\Omega },u=0\text{on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^N$ with $N\ge 3$, $p=(N+2)\mathrm{/}(N-2)$, $1\le q<p$ and $\epsilon$ is a positive parameter. We first determine the limits of the ordered weighted eigenvalues associated with families of positive functions admitting an $m$-bubble decomposition at distinct interior points with a remainder vanishing in $H_0^1(\mathrm{\Omega })$. The limiting eigenvalues are those of the single-bubble problem, with all multiplicities multiplied by $m$. We then refine this analysis for positive multi-bubble solutions of this problem with comparable concentration scales, deriving precise expansions of the first $(N+2)m$ eigenvalues and characterizing their eigenfunctions when $1<q<3$ if $N=4$ or $1\le q<p$ if $N\ge 5$. The leading corrections are determined by explicit symmetric matrices involving $q$, the concentration parameters, and the Green and Robin functions. In particular, we prove that the dilation matrix is positive definite. Together, these spectral results yield Morse index bounds for sufficiently small $\epsilon$, with an exact formula when the matrix governing the translation modes is invertible.