Oscillations of solutions of perturbed autonomous equations with an application to nonlinear elliptic eigenvalue problems involving critical Sobolev exponents

We discuss radially symmetric, not necessarily positive, solutions of the Dirichlet problem for t:J.u + {h(u, r) + >..f2(u, r)} = 0 in the unit ball BN, N > 2, where h and h are suitably homogeneous functions with h critical and f2 subcritical.Estimates are obtained for asymptotic relations between u(O) and >...The method transforms the problem to a study of the zeros of certain solutions of a perturbed autonomous ordinary differential equation.However we do not confine our attention to power-type behaviour in w.We view (1.1) as a perturbation of the autonomous equation w"-w+g(w)=O, and this in turn as a pertubation of w"-w = 0