General-domain multi-peak nodal solutions

Determine general geometric or topological conditions on a smooth bounded domain Ω that guarantee a stable critical set of a positive eigenvalue of the signed Green–Robin interaction matrix M^σ for configurations with k≥3 concentration points, thereby enabling construction of multi-peak nodal solutions without the substantial symmetries used in the paper.

Background

The paper constructs a positive–negative pair in every smooth bounded domain, because the relevant positive eigenvalue has a compact stable set of minimizers. For three or more concentration points, however, the authors require symmetry assumptions to reduce the finite-dimensional problem to a tractable form.

The unresolved issue is to identify domain geometry or topology that guarantees the existence of the required stable critical set for the signed interaction matrix when k≥3, without relying on such symmetry reductions.

References

Can one find general geometric or topological conditions on $\Omega$ ensuring a stable critical set of a positive eigenvalue of $M{\boldsymbol\sigma}$ for $k\ge3$?

— 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 2 (Multi-peak nodal solutions in general domains)

Which geometric assumptions on a general domain allow an analogous $1+k$ construction? Can one construct families for which the difference between the numbers of positive and negative concentration points is larger than one without imposing radial symmetry?

— 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 4 (Asymmetric numbers of positive and negative peaks outside the ball)

To what extent can one classify isolated sign-changing blow-up configurations in dimension four and prove that their limiting parameters must satisfy the spectral criticality conditions appearing in Theorem~\ref{main}?

— 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 7 (A converse blow-up theory for nodal solutions)