Published online by Cambridge University Press: 20 November 2018
We consider the prescribed boundary mean curvature problem in ${{\mathbb{B}}^{N}}$ with the Euclidean metric
where $\tilde{K}\left( x \right)$ is positive and rotationally symmetric on ${{\mathbb{S}}^{N-1}},{{2}^{\#}}=\frac{2\left( N-1 \right)}{N-2}$. We show that if $\tilde{K}\left( x \right)$ has a local maximum point, then this problem has infinitely many positive solutions that are not rotationally symmetric on ${{\mathbb{S}}^{N-1}}$.