A priori estimates of Nodal solutions on the annulus for some PDE and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Delta u + f (u) = 0 in Omega, where Omega is an annulus in R-N, N >= 2, and f satisfies some appropriate conditions. We establish a priori estimates of radial solutions with k prescribed number of zeros. Moreover, when f (r, s) = K(r) g (s), where g is odd and super-linear at infinity, using the uniqueness result due to Tanaka (2007), we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.