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

Abstract

By using a characterization of the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in [4], we give a lower bound for the Morse index of radial solutions to Hénon type problemswhere Ω is a bounded radially symmetric domain of R N (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−1 u, 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.

Morse index computation for radial solutions of the {Hé}non problem in the disk

“…as proved in [17] for the case α = 0 (see also [1,10] for previous results in this direction) and then for the case α > 0 in [8] by exploiting the relation in (1.2) (see also [18]). Here N stands for the dimension, N j = (N +2j−2)(N +j−3)!…”

“…Focusing on radial solutions u p of problem (1.1), it is known, from [28,11] in the case α = 0 and [8] in the case α > 0, that the radial Morse index m rad (u p ) (i.e. the number of the negative eigenvalues of L up in the subspace H 1 0,rad (B) of the radial functions in H 1 0 (B)), coincides with the number m of nodal zones of u p : m rad (u p ) = m and moreover the solution u p is radially nondegenerate.…”

“…m ≥ 2) solution for problem (1.1), having Morse index 2 (cfr. [11]), can not be radial (see [8] and also [1,10,18]).…”

“…where, as in the previous sections, u p denotes the unique radial solution of the Lane-Emden problem with m nodal zones and positive at the origin. The interested reader can find more details in [21,8,29] and the references therein. The strategy summarized in Section 3 applies also to the Hénon problem (see [3]), indeed the Morse index of u α,p is equal to the number of the negative eigenvalues Λ α (p) of (6.3)…”

Global bifurcation for the Hénon problem

“…Coming to nodal solutions, considerations based on the Morse index yield that the minimal energy solution is nonradial for every α ≥ 0. Indeed the minimal energy nodal solution has Morse index 2, while the Morse index of nodal radial solutions is greater, see [6]. Sign-changing multi-bubble solutions have been produced by finitedimensional reduction methods, we can quote [9] and the references therein for the Lane-Emden problem and [25] for the Hénon problem in the disk.…”