Sign-changing solutions for the critical Neumann problem in the three-dimensional unit ball
Abstract
We consider the existence of nonradial sign-changing solutions to the problem Δu−μu+|u|4u=0 with zero Neumann boundary conditions in the unit ball for an arbitrary fixed parameter μ>0. We prove that, for every μ>0, this problem admits infinitely many distinct nonradial sign-changing solutions. More precisely, for every sufficiently large even integer K, we construct a solution by gluing K symmetrically arranged, suitably transformed copies of a sign-changing entire solution of the critical equation in ℝ3. The construction combines an inner--outer gluing scheme with a finite-dimensional reduction. A detailed expansion of the reduced energy allows us to locate an interior critical point in the admissible parameter region and thereby complete the construction.
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BibTeX
@misc{wu2026signchanging,
title = {{Sign-changing solutions for the critical Neumann problem in the three-dimensional unit ball}},
author = {Fei Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02857},
eprint = {2610.02857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02857},
}
RIS
TY - UNPB
TI - Sign-changing solutions for the critical Neumann problem in the three-dimensional unit ball
AU - Fei Wu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02857
UR - https://arxiv.org/abs/2610.02857
AB - We consider the existence of nonradial sign-changing solutions to the problem Δu−μu+|u|^4u=0 with zero Neumann boundary conditions in the unit ball for an arbitrary fixed parameter μ>0. We prove that, for every μ>0, this problem admits infinitely many distinct nonradial sign-changing solutions. More precisely, for every sufficiently large even integer K, we construct a solution by gluing K symmetrically arranged, suitably transformed copies of a sign-changing entire solution of the critical equation in ℝ^3. The construction combines an inner--outer gluing scheme with a finite-dimensional reduction. A detailed expansion of the reduced energy allows us to locate an interior critical point in the admissible parameter region and thereby complete the construction.
ER -