One bubble against a ring: new sign-changing solutions of the Brezis-Nirenberg problem
Abstract
We consider the Brezis--Nirenberg problem −Δu=|u|2*−2u+ε u in the unit ball of ℝN, N≥5, with Dirichlet boundary conditions, and we look for sign-changing solutions which, as ε→0+, blow up positively at the centre and negatively at the k vertices of a regular polygon. The key observation is that, for this configuration, the gradient of the reduced energy factorises, so that the whole finite-dimensional problem is governed by the zeros of a single explicit function of the radius of the polygon. Its behaviour depends on the dimension: when N=5 such solutions exist for every k≥2, and the ring of negative peaks approaches the boundary at distance O(1/k); when N≥6 there are two solutions for 2≤ k≤6, while no solution of this type exists for k large. For 2≤ k≤6 the nodal set of every such solution reaches the boundary of the ball, in contrast with what happens for the almost critical problem. Nevertheless, the two solutions found for N≥6 have very different nodal structures: numerically, for 3≤ k≤6 one of them has only two nodal domains, and its nodal set meets the plane of the peaks along a closed curve around the centre, while the other one can have as many as k+1 nodal domains. These results extend to higher dimensions a configuration previously found in dimension four.
Related
- Spectral analysis of multi-bubble solutions to the Brezis-Nirenberg problem
- Sign-changing bubbling solutions on polygons for the Brezis-Nirenberg problem in high dimensions
- Sign-changing solutions for the critical Neumann problem in the three-dimensional unit ball
- Spectral Characterization, Local Rigidity and Morse Index of Bubble Solutions to the CR Fractional Yamabe Problem
- The CR Nirenberg Problem, Part I: Blow-up Analysis and Compactness
BibTeX
@misc{pistoia2026bubble,
title = {{One bubble against a ring: new sign-changing solutions of the Brezis-Nirenberg problem}},
author = {Angela Pistoia and Giuseppe Mario Rago and Giusi Vaira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26107},
eprint = {2610.26107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26107},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One bubble against a ring: new sign-changing solutions of the Brezis-Nirenberg problem
AU - Angela Pistoia
AU - Giuseppe Mario Rago
AU - Giusi Vaira
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26107
UR - https://arxiv.org/abs/2610.26107
AB - We consider the Brezis--Nirenberg problem −Δu=|u|^2^*−2u+ε u in the unit ball of ℝ^N, N≥5, with Dirichlet boundary conditions, and we look for sign-changing solutions which, as ε→0^+, blow up positively at the centre and negatively at the k vertices of a regular polygon. The key observation is that, for this configuration, the gradient of the reduced energy factorises, so that the whole finite-dimensional problem is governed by the zeros of a single explicit function of the radius of the polygon. Its behaviour depends on the dimension: when N=5 such solutions exist for every k≥2, and the ring of negative peaks approaches the boundary at distance O(1/k); when N≥6 there are two solutions for 2≤ k≤6, while no solution of this type exists for k large. For 2≤ k≤6 the nodal set of every such solution reaches the boundary of the ball, in contrast with what happens for the almost critical problem. Nevertheless, the two solutions found for N≥6 have very different nodal structures: numerically, for 3≤ k≤6 one of them has only two nodal domains, and its nodal set meets the plane of the peaks along a closed curve around the centre, while the other one can have as many as k+1 nodal domains. These results extend to higher dimensions a configuration previously found in dimension four.
ER -