Infinitely many solutions for the prescribed boundary mean curvature problem on $\mathbb B^N$

TL;DR

This paper proves the existence of infinitely many positive, non-radial solutions to a boundary mean curvature problem on the unit ball in N-dimensional space, under conditions on the prescribed curvature function.

Contribution

It establishes the existence of infinitely many solutions for the boundary mean curvature problem with a positive, rotationally symmetric curvature function having a local maximum.

Findings

Infinitely many positive solutions exist.

Solutions are non-radial on the sphere.

Results depend on the curvature function having a local maximum.

Abstract

We consider the following prescribed boundary mean curvature problem in $\mathbb B^N$ with the Euclidean metric $-\Delta u =0$, $u>0$ in $B^N, \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} K(x) u^{N/(N-2)}$ on $S^{N-1}, where $K$ is positive and rotationally symmetric on $\mathbb S^{N-1}$. We show that if ${K}$ has a local maximum point, then the equation has {\bf infinitely many positive} solutions, which are non-radial on $\mathbb S^{N-1}$.