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

Baishun Lai; Qing Luo

doi:10.3934/dcds.2011.30.227

Article Contents

2011, Volume 30, Issue 1: 227-241. Doi: 10.3934/dcds.2011.30.227

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

Baishun Lai^1, and
Qing Luo^2,

1.
Institute of Contemporary Mathematics, Henan University, School of Mathematics and Information Science, Henan University, Kaifeng 475004
2.
School of Mathematics and Information Science, Henan University, Kaifeng 475004

Received: December 31, 2009

Revised: April 30, 2010

Published: December 2010

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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.

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Mathematics Subject Classification: Primary 35B45; Secondary 35J40.

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References

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