`a`
Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity

Pages: 227 - 241, Volume 30, Issue 1, May 2011      doi:10.3934/dcds.2011.30.227

 
       Abstract        References        Full Text (402.2K)       Related Articles

Baishun Lai - Institute of Contemporary Mathematics, Henan University, School of Mathematics and Information Science, Henan University, Kaifeng 475004, China (email)
Qing Luo - School of Mathematics and Information Science, Henan University, Kaifeng 475004, China (email)

Abstract: In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation

$\beta \Delta^{2}u-\tau \Delta u=\frac{\lambda}{(1-u)^{p}} \mbox{in} B$,
$0 < u \leq 1 \mbox{in} B$,
$u=\Delta u=0 \mbox{on} \partial B$,

where $B$ is the unit ball in $R^{n}$, $\lambda>0$ is a parameter, $\tau>0, \beta>0,p>1$ are fixed constants. By Hardy-Rellich inequality, we find that when $p$ is large enough, the critical dimension is 13.

Keywords:  Minimal solutions, regularity, critical dimension, stability, fourth order.
Mathematics Subject Classification:  Primary 35B45; Secondary 35J40.

Received: January 2010;      Revised: May 2010;      Available Online: February 2011.

 References