On stable solutions of the biharmonic problem with polynomial growth

Hatem Hajlaoui, Abdellaziz Harrabi and Dong Ye

Vol. 270 (2014), No. 1, 79–93

DOI: 10.2140/pjm.2014.270.79

Abstract

We prove the nonexistence of smooth stable solutions to the biharmonic problem ${\Delta }^{2}u=u^p$, $u>0$ in ${ℝ}^N$ for $1<p<\infty$ and $N<2\left(1+x_0\right)$, where $x_0$ is the largest root of the equation

$$x^4-\frac{32p\left(p+1\right)}{{\left(p-1\right)}^2}{x}^2+\frac{32p\left(p+1\right)\left(p+3\right)}{{\left(p-1\right)}^3}x-\frac{64p{\left(p+1\right)}^2}{{\left(p-1\right)}^4}=0.$$

In particular, as $x_0>5$ when $p>1$, we obtain the nonexistence of smooth stable solutions for any $N\le 12$ and $p>1$. Moreover, we consider also the corresponding problem in the half-space ${ℝ}_{+}^N$, and the elliptic problem ${\Delta }^{2}u=\lambda {\left(u+1\right)}^p$ on a bounded smooth domain $\Omega$ 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