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

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].