Morse index and uniqueness for positive solutions of radial p-Laplace equations

We study the positive radial solutions of the Dirichlet problem Delta(p)u+ f( u) = 0 in B, u > 0 in B, u = 0 on partial derivativeB, where Delta(p)u = div(delu(p-2) delu), p > 1, is the p-Laplace operator, B is the unit ball in R-n centered at the origin and f is a C-1 function. We are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the Morse index. In particular, we show that positive radial solutions of Mountain Pass type have Morse index 1 in the subspace of radial functions of W-0(1,p) (B). We use this to prove uniqueness and nondegeneracy of positive radial solutions when f is of the type u(s) + u(q) and p greater than or equal to 2.