Emden equation involving the critical Sobolev exponent with the third-kind boundary condition in S³

Abstract

We consider a positive solution of the Emden equation with the critical Sobolev exponent on a geodesic ball in S³. In the case of the Dirichlet boundary condition, Bandle and Peletier [2] proved the precise result on the existence of a positive radial solution. We investigate the same equation with the third kind boundary condition and obtain a more general result. Namely we prove that the existence and the nonexistence of solutions depend on the geodesic radius and the boundary condition. Moreover the set of solutions consists of a unique radial classical solution and a continuum of singular solutions.