
 
Singularity of the Extremal Solution for SupercriticalBiharmonic Equations with PowerType Nonlinearity 
 
Citation： 
Baishun LAI,Zhengxiang YAN,Yinghui ZHANG.Singularity of the Extremal Solution for SupercriticalBiharmonic Equations with PowerType Nonlinearity[J].Chinese Annals of Mathematics B,2017,38(3):815~826 
Page view： 1334
Net amount： 1061 
Authors： 
Baishun LAI; Zhengxiang YAN;Yinghui ZHANG 
Foundation： 
This work was supported by the National Natural Science
Foundation of China (Nos.11201119, 11471099), the International
Cultivation of Henan Advanced Talents and the Research Foundation of
Henan University (No.yqpy20140043). 


Abstract： 
Let $\B\subset \R^{n}$ be the unit ball centered at the origin. The
authors consider the following biharmonic equation:
$$
\left\{\!\!\!
\begin{array}{lllllll}
\Delta^{2}u=\lambda(1+u)^{p} & \mbox{in}\ \B, \u=\ds\frac{\partial u}{\partial \nu} =0 & \mbox{on}\ \partial \B,\\end{array}
\right.
$$
where $p>\frac{n+4}{n4}$ and $\nu$ is the outward unit normal
vector. It is wellknown that there exists a $\lambda^{*}>0$ such
that the
biharmonic equation has a solution for $\lambda\in(0,\lambda^{*})$ and has a
unique weak solution $u^{*}$ with parameter $\lambda=\lambda^{*}$, called the
extremal solution. It is proved that $u^{*}$ is singular when
$n\geq 13$ for $p$ large enough and satisfies $u^{*}\leq
r^{\frac{4}{p1}}1$ on the unit ball, which actually solve a part
of the open problem left in [D\`{a}vila, J., Flores, I., Guerra,
I., Multiplicity of solutions for a fourth order equation with
powertype nonlinearity, {\it Math. Ann.}, {\bf 348}(1), 2009,
143193]. 
Keywords： 
Minimal solutions, Regularity, Stability, Fourth order 
Classification： 
35B45, 35J40 

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