Vol. 270, No. 1, 2014

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ISSN: 0030-8730
On stable solutions of the biharmonic problem with polynomial growth

Hatem Hajlaoui, Abdellaziz Harrabi and Dong Ye

Vol. 270 (2014), No. 1, 79–93
Abstract

We prove the nonexistence of smooth stable solutions to the biharmonic problem Δ2u = up, u > 0 in N for 1 < p < and N < 2(1 + x0), where x0 is the largest root of the equation

x4 32p(p + 1) (p 1)2 x2 + 32p(p + 1)(p + 3) (p 1)3 x 64p(p + 1)2 (p 1)4 = 0.

In particular, as x0 > 5 when p > 1, we obtain the nonexistence of smooth stable solutions for any N 12 and p > 1. Moreover, we consider also the corresponding problem in the half-space +N, and the elliptic problem Δ2u = λ(u + 1)p on a bounded smooth domain Ω with the Navier boundary conditions. We prove the regularity of the extremal solution in lower dimensions.

Keywords
stable solutions, biharmonic equations, polynomial growths
Mathematical Subject Classification 2010
Primary: 35J91
Secondary: 35J30, 35J40
Milestones
Received: 1 March 2013
Accepted: 7 November 2013
Published: 2 August 2014
Authors
Hatem Hajlaoui
Institut de Mathématiques Appliquées et d’Informatiques
3100 Kairouan
Tunisia
Abdellaziz Harrabi
Institut de Mathématiques Appliquées et d’Informatiques
3100 Kairouan
Tunisia
Dong Ye
IECL, UMR 7502
Université de Lorraine
57045 Metz
France