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| Singularity of the Extremal Solution for SupercriticalBiharmonic Equations with Power-Type Nonlinearity |
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Citation： |
Baishun LAI,Zhengxiang YAN,Yinghui ZHANG.Singularity of the Extremal Solution for SupercriticalBiharmonic Equations with Power-Type Nonlinearity[J].Chinese Annals of Mathematics B,2017,38(3):815~826 |
| Page view： 3957
Net amount： 4225 |
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). |
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| 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 $p>\frac{n+4}{n-4}$ and $\nu$ is the outward unit normal
vector. It is well-known 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 $u^{*}\leq
r^{-\frac{4}{p-1}}-1$ 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
power-type nonlinearity, {\it Math. Ann.}, {\bf 348}(1), 2009,
143--193]. |
Keywords： |
Minimal solutions, Regularity, Stability, Fourth order |
Classification： |
35B45, 35J40 |
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