Sign-changing bubbling solutions on polygons for the Brezis-Nirenberg problem in high dimensions
Abstract
We consider the Brezis--Nirenberg problem −Δu = |u|2*−2u + ε u in a smooth bounded domain Ω⊂ ℝN, N ≥ 5, with Dirichlet boundary conditions, where ε > 0 is small. Two multi-peak configurations found in dimension four in \cite{PRV2} persist. In an annulus, for every even k ≥ 2, we construct solutions blowing up at the vertices of a regular k-gon with alternating signs. In the ball, we construct solutions blowing up positively at the vertices of a regular k-gon in a coordinate plane and negatively at those of a regular k-gon in an orthogonal plane, for 2 ≤ k ≤ 5 if N = 5,6 and 2 ≤ k ≤ 4 if N ≥ 7. This range is sharp: for larger k no such solution exists with parameters in a compact set, while otherwise the reduced energy has a unique, non-degenerate critical point.
Related
- One bubble against a ring: new sign-changing solutions of the Brezis-Nirenberg problem
- Unique continuation, nonexistence and bubbling for the critical p-Laplace equation II: the degenerate elliptic case and higher dimensions
- Non-degeneracy and local uniqueness of bubbling solutions for a singular Liouville equation
- Critical functions and asymptotics for fractional Hartree Brezis--Nirenberg problems in low dimensions
- Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations
BibTeX
@misc{rago2026signchanging,
title = {{Sign-changing bubbling solutions on polygons for the Brezis-Nirenberg problem in high dimensions}},
author = {Giuseppe Mario Rago},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26180},
eprint = {2610.26180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26180},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sign-changing bubbling solutions on polygons for the Brezis-Nirenberg problem in high dimensions
AU - Giuseppe Mario Rago
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26180
UR - https://arxiv.org/abs/2610.26180
AB - We consider the Brezis--Nirenberg problem −Δu = |u|^2^*−2u + ε u in a smooth bounded domain Ω⊂ ℝ^N, N ≥ 5, with Dirichlet boundary conditions, where ε > 0 is small. Two multi-peak configurations found in dimension four in \cite{PRV2} persist. In an annulus, for every even k ≥ 2, we construct solutions blowing up at the vertices of a regular k-gon with alternating signs. In the ball, we construct solutions blowing up positively at the vertices of a regular k-gon in a coordinate plane and negatively at those of a regular k-gon in an orthogonal plane, for 2 ≤ k ≤ 5 if N = 5,6 and 2 ≤ k ≤ 4 if N ≥ 7. This range is sharp: for larger k no such solution exists with parameters in a compact set, while otherwise the reduced energy has a unique, non-degenerate critical point.
ER -