Extension to the full exponent range above the Sobolev threshold

Establish whether concentration near arbitrary nondegenerate critical points of Q persists for every exponent range n/(n−2)<q<p<(n+2)/(n−2), rather than only for q>q₁.

Background

The main theorem assumes q>q₁, where q₁ is a dimension-dependent threshold strictly larger than n/(n−2). The paper therefore does not cover the larger interval n/(n−2)<q≤q₁. The unresolved issue is whether arbitrary prescribed nondegenerate critical points of the potential remain valid concentration locations throughout that broader exponent range.

References

Several natural questions remain open. First, our argument requires the dimensional restriction n\geq9. This condition enters through the estimates of the algebraic interactions and of the error terms, and it would be interesting to understand whether it is merely technical or whether a different phenomenon occurs in lower dimensions. Second, the present construction requires q to lie above the threshold q_1>n/(n-2). It is natural to ask whether concentration near arbitrary nondegenerate critical points persists throughout the larger range \frac{n}{n-2}<q<p.

— Existence of multi-peak solutions for singularly perturbed problems in the zero-mass case  (2609.03135 - Demuth, 2 Sep 2026) in Section 1, Introduction