Solutions of semilinear elliptic problems in shrinking annuli

Abstract

The positive, radially symmetric solutions of semilinear Dirichlet problems in annuli is studied, the inner radius of which is sufficiently small. By means of a phase plane analysis their asymptotic behaviour is computed as the inner radius shrinks. It is of particular interest in the cases where the Dirichlet problem for the sphere has no non-trivial solution. This work extends results of Bandle and Peletier [ Math. Ann. 280 (1988), 1–19].