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arXiv · 2610.02857

Sign-changing solutions for the critical Neumann problem in the three-dimensional unit ball

Fei Wu
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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 $\mathbb{R}^3$. 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.

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Fei Wu. 2026-10-02. Sign-changing solutions for the critical Neumann problem in the three-dimensional unit ball. https://arxiv.org/abs/2610.02857

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