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Several problems for the differential equation Lu=g(r,u) with Lpu=r-(rlu'lP-2u') are considered. For ot N-1, the operator L is the radially symmetric p-Laplacian in JR". For various uniqueness conditions the initial value problem with given data u(ro) uo u' (ro) u and counterexamples to uniqueness are given. For the case where g is increasing in u, a sharp comparison theorem is established; it leads to maximal solutions, nonuniqueness and uniqueness results, among others. Using these results, a strong comparison principle for the boundary value problem and a number of properties of blow-up solutions are proved under weak assumptions on the nonlinearity g(r, u).

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