Persistence of multi-peak configurations in higher dimensions

Prove that all sign-changing multi-peak configurations constructed for the four-dimensional Brezis–Nirenberg problem— including the five-point aligned configuration and the asymmetric one-positive-plus-k-negative pattern—persist for dimensions N≥5 by analyzing the coupled finite-dimensional reduced functional.

Background

The paper constructs several sign-changing multi-bubble solutions for the four-dimensional Brezis–Nirenberg problem using a spectral reduction involving a signed Green–Robin interaction matrix. In dimension four, the exponentially small concentration scales can be determined through an eigenvalue problem, leaving a location problem for a stable critical set.

For dimensions N≥5, the corresponding reduced functional depends simultaneously on the concentration locations and individual scales. The authors explicitly conjecture that the configurations obtained in dimension four persist in higher dimensions, but the coupled critical-point analysis remains unresolved.

References

We strongly conjecture that all the multi-peak configurations constructed in Theorem~\ref{main-configurations} persist in dimensions $N\ge5$. The main obstacle is that the scale variables no longer reduce to a spectral problem. Instead one has to analyze the full finite-dimensional functional signed-reduced-functional, where the locations and the relative scales remain coupled. Proving this conjecture, even for the five-point aligned configuration or for the asymmetric $1+k$ pattern, would already be of interest.

signed-reduced-functional:

Fσ(d,ξ)=aN⟨Mσ(ξ)d,d⟩−bN∑j=1kdj4N−2,F^{\boldsymbol\sigma}(\boldsymbol d,\boldsymbol\xi) = a_N\left\langle M^{\boldsymbol\sigma}(\boldsymbol\xi)\boldsymbol d, \boldsymbol d \right\rangle - b_N\sum_{j=1}^kd_j^{\frac{4}{N-2}},

— 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 1 (Higher dimensions)

Can one construct aligned alternating configurations with seven or more peaks? More generally, is there a systematic spectral criterion for the symmetry-reduced interaction matrices associated with an arbitrary odd number of aligned peaks?

— 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 3 (More aligned peaks)