Radial solutions of an elliptic equation with singular nonlinearity

For the equation -Delta u + u(-beta) = u(p), u > 0 in B-R, u = 0 on partial derivative B-R, where B-R subset of R-N, 0 < beta < 1 and 1 < p < N+2/N-2 if N >= 3, 1 < p < +infinity if N = 2, we show that there is (R) over bar > 0 such that a radial solution u(R) exists if and only if 0 < R <= <(R)over bar>. It is unique in the class of radial solutions and u(R)' (R) < 0 if R < (R) over bar, while u((R) over bar)'((R) over bar) = 0. We also give a variational characterization of u((R) over bar). (C) 2008 Elsevier Inc. All rights reserved.