On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE’s

Abstract We investigate nodal radial solutions to semilinear problems of type − Δ u = f ( | x | , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded radially symmetric domain of R N ( N ≥ 2 ) and f is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse index in a forthcoming work.