Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

Abstract

We examine the fourth order problem ∆ 2 u = λf (u) in Ω with ∆u = u = 0 on ∂Ω, where λ > 0 is a parameter, Ω is a bounded domain in R N and where f is one of the following nonlinearities: f (u) = e u , f (u) = (1 + u) p or f (u) = 2010 Mathematics Subject Classification.

Liouville type results for semi-stable solutions of the weighted Lane–Emden system

“…A new approach which is obtained by Cowan-Ghoussoub [2] and Dupaigne-Ghergu-Goubet-Warnault [7] independently can handle the Lane-Emden system and biharmonic equation with negative exponent. Thus one combine the second order stability criterion with bootstrap iteration.…”

Liouville theorems for stable solutions of the weighted Lane-Emden system

“…• Adopting the new approach in [5], Hu proved the following Liouville theorem for classical stable solutions of (1.1) for ρ = ρ 0 and θ ≥ p ≥ 2 or θ = p > 4 3 , obtaining a direct extension of Theorem 1 in [2] for ρ ≡ 1. More precisely, let t + 0 and t − 0 be the quantities used in [2] :…”

Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

“…In particular, when p = 2, u * is bounded away from 1 for n ≤ 5. The later result (and also the general case 1 < p = 3) is improved in [7] to n ≤ 6, and further improved by Guo and Wei in [14] to n ≤ 7. However, for p = 2 the expected optimal dimension is n = 8, holds on the ball, see [15].…”

“…The function λ → u λ is strictly increasing on (0, λ * ), the increasing pointwise limit u * (x) = lim λ↑λ * u λ (x) is called the extremal solution. For 0 < λ < λ * the minimal solution u λ of problem (N λ ) satisfies the following stability inequality, for the proof see Corollary 1 in [7] or Lemma 6.1 in [11],…”

“…As we shall see, in this paper we prove the same for a large class of nonlinearities including e u . When f (u) = (1 + u) p (p > 1) in [6] it is proved that u * is bounded if n < 8p p−1 that improved in [7] for to n < 4h(p) > 8p p−1 (for the definition of h(p) which is a 2 decreasing function on (1, ∞) see [7]) with lim p→∞ 4h(p) ≈ 10.718. Recently, Hajlaoui, Harrabi and Ye in [18] improved this result by showing that u * is bounded for any p > 1 and n ≤ 12.…”