Symmetry and nonexistence results for Emden-Fowler equations in cones. (English) Zbl 1056.35062

Summary: 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 [Arch. Ration. Mech. Anal. 112, No. 4, 319–338 (1990; Zbl 0727.35051)]. For this one needs to establish some asymptotics for the solutions as \(r\to 0\) or \(r\to\infty\), relying on methods used by M.-F. Bidaut-Veron and L. Veron in [Invent. Math. 106, No. 3, 489–539 (1991; Zbl 0755.35036)] 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 [P. Padilla, Appl. Anal. 64, No. 1–2, 153–169 (1997; Zbl 0942.35084)].


35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50 Maximum principles in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs