Spectral theory for linearized pp-Laplace equations

We continue and completely set up the spectral theory initiated in Castorina et al. (2009) [5] for the linearized operator arising from Δpu+f(u)=0Δpu+f(u)=0. We establish existence and variational characterization of all the eigenvalues, and by a weak Harnack inequality we deduce Hölder continuity for the corresponding eigenfunctions, this regularity being sharp. The Morse index of a positive solution can be now defined in the classical way, and we will illustrate some qualitative consequences one should expect to deduce from such information. In particular, we show that zero Morse index (or more generally, non-degenerate) solutions on the annulus are radial.