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

Global bifurcation for the Hénon problem

“…. m and j such that the related spherical harmonic belongs to V , by the characterization in [5,Theorem 1.4]. Taking advantage from the description of the spherical harmonics given in the proof of Theorem 1.1 in [4], one sees that j must be a multiple of n and so, in particular,…”

“…the number of the negative eigenvalues of for (3.3) whose relative eigenfunction is H 1 0,rad (B), the subspace of H 1 0 (B) given by radial functions. As explained in full details in [5], this matter can be regarded through a singular eigenvalue problem associated to the linearized operator L p , which has to be handled in weighted Lebesgue and Sobolev spaces…”

“…see [5,6]. In both cases M is a real parameter given by L q M = {ϕ : (0, 1) → R : ϕ measurable and s.t.…”

“…5) holds. Thanks to(3.19) and(3.26), it means that−β 2 = lim p→1 ν 1 (p) > − 2n 2 lim p→∞ ν 1 (p) = −κ 2 , which is clearly equivalent to 2+α 2 β < n < 2+α 2 κ, i.e.…”

“…This approach has already been applied to the Lane-Emden problem in an annulus, see [15], and then extended to higher dimension and to sign-changing solutions in [4]. It can be applied also to the Hénon equation because the exact computations in [8,7,2] rely on a characterization of the Morse index in terms of a singular Sturm-Liouville problem from [5], which allows to describe in full details the kernel of the linearized operator. Furthermore this tool provides a detailed bifurcation analysis also for positive solutions, and in the subcritical case, since it gives informations about the symmetries of the bifurcating solutions and the global properties of the branches.…”