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 typewhere 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, and describe the symmetries of the eigenfunctions. Next we use this characterization to give a lower bound for the Morse index; in such a way we give an alternative proof of an already known estimate for the autonomous problem and we furnish a new estimate for Hénon type problems with f (|x|, u) = |x| α f (u). Concerning the real Hénon problem, f (|x|, u) = |x| α |u| p−1 u, we prove radial nondegeneracy and show that the radial Morse index is equal to the number of nodal zones.

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

“…In this section we give all the notations we need in the following, we introduce the singular eigenvalue problems that have been the subject of [4] and we recall their relation with the Morse index of a solution u to (1.1) that we need to prove the main results. Since this paper is the sequel of [4] we suggest to read the first part where some properties of the singular eigenvalues and eigenfunctions are proved. In the following Ω denotes a bounded radially symmetric domain of R N , while B = {x ∈ R N : |x| < 1} is the unit ball.…”

“…. Following [4] we use some singular eigenvalues associated to the linearized operator L u to characterize the Morse index of a solution u to (1.1). To define them we need some weighted Lebesgue and Sobolev spaces that we denote by L := {ψ : Ω → R : ψ measurable and s.tˆΩ |x| −2 ψ 2 dx < ∞},…”

“…The proof of Theorem 1.1 relies on a transformation of the radial variable which, like the one in [21], brings radial solutions to problem (1.1) into solutions of a suitable autonomous o.d.e (see [4,Sect. 4.1]).…”

“…4.1]). The main difference in our approach is that we compute the Morse index starting from a singular eigenvalue problem studied in the the first part of this paper, [4]. In that way the core of the proof stands in an estimate of the singular eigenvalues given in Proposition 3.3.…”

“…In that way the core of the proof stands in an estimate of the singular eigenvalues given in Proposition 3.3. Such estimate, together with [4,Corollary 4.11], allows to obtain informations also on the Morse index in symmetric spaces and has interesting implications on the multiplicity of solutions, as discussed with more details at the end of Section 4.…”