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

We consider the existence of nonradial sign-changing solutions to the problem $\mathrm{\Delta }u-\mu u+\mathrm{\mid }u{\mathrm{\mid }}^{4}u=0$ with zero Neumann boundary conditions in the unit ball for an arbitrary fixed parameter $\mu >0$. We prove that, for every $\mu >0$, 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.