Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

Let ${\lambda }^{\ast }>0$ denote the largest possible value of $\lambda$ such that \{{array}{lllllll} \Delta^{2}u=\lambda(1+u)^{p} & {in}\ \ \B, %0<u\leq 1 & {in}\ \ \B, u=\frac{\partial u}{\partial n} =0 & {on}\ \ \partial \B {array}. has a solution, where $\text{B}$ is the unit ball in $R^n$ centered at the origin, $p>\frac{n+4}{n-4}$ and $n$ is the exterior unit normal vector. We show that for $\lambda ={\lambda }^{\ast }$ this problem possesses a unique weak solution $u^{\ast }$, called the extremal solution. We prove that $u^{\ast }$ is singular when $n\ge 13$ for $p$ large enough, in which case $u^{*}(x)\leq r^{-\frac{4}{p-1}}-1$ on the unit ball.