Singularity of the Extremal Solution for SupercriticalBiharmonic Equations with Power-Type Nonlinearity


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
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Baishun LAI; Zhengxiang YAN;Yinghui ZHANG


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 $p>\frac{n+4}{n-4}$ 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].


Minimal solutions, Regularity, Stability, Fourth order


35B45, 35J40
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