The Hénon problem with large exponent in the disc

Abstract

In this paper we consider the Hénon problem in the unit disc with Dirichlet boundary conditions. We study the asymptotic profile of least energy and nodal least energy radial solutions and then deduce the exact computation of their Morse index for large values of the exponent p. As a consequence of this computation a multiplicity result for positive and nodal solutions is obtained.

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, .…”

Global bifurcation for the Hénon problem

“…After multi-peak solutions have been constructed by finite-dimensional reduction methods under various incidental assumptions, we mention [16,23] among others. Nonradial solutions have also been produced by variational methods as in [8,1], after imposing some constrains on the symmetries of the solutions, and by bifurcation methods in [3,17].…”

“…Some of these quasi-radial solutions are produced as least energy nodal solutions in symmetric spaces, some others by bifurcation w.r.t. the parameter p. The approach of least energy solutions in symmetric space has been extended also to the Hénon equation in [1,2], always in dimension N = 2. Concerning the Hénon equation in dimension N ≥ 3, in the subcritical case a very recent paper by Kübler and Weth [21] produced an infinite number of nonradial solutions by bifurcation w.r.t.…”

“…the parameter p ∈ (1, p α ), for any given value of α > 0 (and also α = 0, as far as sign-changing solutions are concerned), so we must take into account also the supercritical case. The Morse index of radial solutions when the parameter p approaches the supremum of the existence range has been recently computed in four different papers ( [15,14] concerning the Lane-Emden problem in dimension N = 2 and N ≥ 3 respectively, and [1,5] for the Hénon problem), while when p is close to 1 it has been characterized in terms of the zeros of suitable Bessel functions in [2]. Starting from these computations we see that for the positive solution to the Hénon equation the Morse index for p close to 1 is lower than at the supremum of the existence range, and the same holds for nodal solutions in dimension N = 2, while in dimension N ≥ 3 the inequality is reversed.…”

“…In the disc solutions enjoying the same symmetry properties have been produced in [16] by the Lyapunov-Schmidt reduction method, and in [1] by minimizing the energy associated to (1.1) in the space H 1 0,n . In this last paper it has been proved that such "least energy n-invariant solutions"are nonradial and different one from another at least for p ∈ (p n , +∞), with p n the same exponent appearing here.…”