Asymptotic behavior of least energy solutions for critical Neumann systems as λ → ∞

Determine the asymptotic behavior, as λ → ∞, of the least energy solutions constructed in Mauro–Schiera–Tavares (arXiv:2509.18835, Theorem 1.5) for critical Neumann Schrödinger systems in both cooperative (β < 0) and competitive (β > 0) regimes; specifically, ascertain whether the components blow up and, if they do, whether their concentration points coincide (collapse) or remain distinct.

Background

Recent work (cited as (Mauro et al., 23 Sep 2025) establishes existence of least energy solutions for coupled critical Neumann systems in cooperative and competitive regimes. However, the qualitative behavior of these minimizers as the parameter λ becomes large has not been characterized.

The present paper constructs blowing-up solutions under geometric hypotheses on the mean curvature, raising the broader question of whether least energy states exhibit blow-up and, in the competitive case, whether their concentration points remain separated or merge as λ → ∞.

References

It is an interesting open question to study the asymptotic behavior of the least energy solutions found in Theorem 1.5 as \lambda \to \infty. Specifically, it remains to be determined whether the components blow up, and if so, whether their concentration points collapse or remain distinct.

— Blowing-up solutions to a critical 4D Neumann system in a competitive regime  (2603.29329 - Guo et al., 31 Mar 2026) in Remark, end of Section 1 (Introduction)