Analogue of Gidas-Ni-Nirenberg result in hyperbolic space and sphere

Abstract

Let $u\epsilon C^{2}\left(\overline{\Omega}\right)$be a positive solution of the differential equation $\Delta u+f\left(u\right)=0$ in $\Omega$ with boundary condition u=0 on $\partial\Omega$ where f is a C$^{1}$ function and $\Omega$ is a geodesic ball in the hyperbolic space $\mathbf{H}^{\mathbf{n}}$ $\left(\textrm{respectively}\:\textrm{sphere}\:\mathbf{S^{\mathbf{n}}}\right)$. Further in case of sphere we assume that $\overline{\Omega}$ is contained in a hemisphere. Then we prove that u is radially symmetric.