Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem

Abstract

We compute the Morse index of nodal radial solutions to the Hénon problem −∆u = |x| α |u| p−1 u in B, u = 0 on ∂B, 1

The Hénon problem with large exponent in the disc

“…In the case α = 0, Theorem 1.4 gives back the Morse index of the Lane-Emden problem computed in [DIP] since 2⌈κ⌉ + 2⌈0⌉ = 12. Formula (1.18) highlights a discontinuity of the solution's set of the Hénon problem (1.1) corresponding to the even values of α which is typical of the nonlinear term |x| α and has been already observed in several papers among which we can quote [PT], [GGN], [AG3] as an example. In particular in (1.18) 1 is the amount of the radial Morse index of u 1 p while 2 α 2 is the contribution of the non radial Morse index, due to the term |x| α , and comes from the asymptotic profile in (1.11).…”

“…In [DIP,Lemma 6.7 and 6.9] (see also [AG3,Lemma 2.11]) it is proved that the function f p (r) has an unique maximum point in each nodal zone of v p , precisely there are 0 < c p < t 1,p < d p < 1 such that f p is strictly increasing in (0, c p ) and in (t 1,p , d p ), while it is strictly decreasing in (c p , t 1,p ) and in (d p , 1). Further the convergence of g p to g in C 0 loc [0, ∞) implies that c p ∈ [0, ε 1 p R] if p is large enough, as well as the convergence of h p to h in C 0 loc (0, ∞) implies that d p ∈ [ ε 2 p K , ε 2 p K].…”

“…It is known indeed that also solutions which minimize the energy are not radial when α is large enough, see [SSW]. In this paper, carrying on the study of the Hénon problem started in [AG2,AG3], we consider (1.1) in the unit disc, where it admits solutions for every p > 1 and no critical exponent appears and in particular we focus on large values of p, where concentration phenomena take place and nonradial positive solutions arise. Indeed it has been shown in [AG] and [GGP2], both dealing with the Lane Emden problem, that radial solutions behave like a spike.…”

“…What happens is that the behaviour in the second nodal zone, where the solution is smaller, has a greater influence due to the effect of the singular term in (1.8), and we will see in Section 3 that it gives rise to a multiplicity result. The existence of this sequence α n seems a new phenomenon which is peculiar of dimension 2 since it does not appear in higher dimensions where each nodal region brings the same contribution (radial and nonradial) to the total Morse index, see [AG3]. It suggests that the set of solutions to (1.1) changes in correspondence of that values of α n , and indeed the number of distinct nonradial solutions that we produce later on in Theorem 1.6 increases by one unit.…”

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

“…The previous inequalities will play a role in the proof of some asymptotic results on the Morse index of radial solutions to (3.1) in [5,6]. Now the statement of Theorem for i = 2, .…”