Symmetry and nonexistence results for Emden-Fowler equations in cones

Abstract

The purpose of this paper is to state some qualitative properties of the solutions to the Emden-Fowler equation $\Delta u + r^\sigma u^p = 0$ in a cone with Dirichlet boundary conditions. Namely one can show that every solution has the same symmetry as the cone in some sense; furthermore it is possible to extend the nonexistence results for regular solutions to this equation already stated by C. Bandle and M. Essen in [2]. For this one needs to establish some asymptotics for the solutions as $r\rightarrow 0$ or $r\rightarrow \infty$, relying on methods used by Veron in [25] for similar equations, but in different geometries, and then use a special form of the moving-planes method on a sphere in the spirit of [22].