On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE’s: II**This work was supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Abstract

By using a characterization of the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in Amadori A L and Gladiali F (2018 arXiv:1805.04321), we give a lower bound for the Morse index of radial solutions to Hénon type problems where Ω is a bounded radially symmetric domain of (N ⩾ 2), α > 0 and f is a real function. From this estimate we get that the Morse index of nodal radial solutions to this problem goes to ∞ as α → ∞. Concerning the real Hénon problem, f(u) = |u|p−1u, we prove radial nondegeneracy, we show that the radial Morse index is equal to the number of nodal zones and we get that a least energy nodal solution is not radial.