Uniqueness and asymptotic selection of minimizing configurations

Determine assumptions guaranteeing that the global minimizer of the reduced eigenvalue is unique and nondegenerate, and, for alternating regular k-gons in an annulus, identify which boundary component is selected by minimizing radii ρ_k and derive the precise asymptotic law for ρ_k as k→∞.

Background

Several existence results in the paper use the complete compact set of global minimizers of a symmetry-reduced eigenvalue rather than a uniquely determined critical point. Consequently, the limiting concentration configuration may not be unique or nondegenerate.

For alternating polygons in an annulus, the paper proves only that any sequence of minimizing radii approaches one of the two boundary components as the number of peaks tends to infinity. It does not determine whether the inner or outer component is selected or establish the rate of convergence.

References

Under what assumptions is this minimizer unique and nondegenerate? In the annulus, if $\rho_k$ denotes a minimizing radius for the alternating $k$-gon, we only prove $\operatorname{dist}(\rho_k,{a,1})\to0$ as $k\to\infty$. Which boundary component is selected, and what is the precise asymptotic law for $\rho_k$?

— Sign-changing multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions  (2608.21239 - Pistoia et al., 21 Aug 2026) in Section 1, subsection “Open problems and further directions,” item 6 (Selection and uniqueness of the limiting configuration)