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

Abstract

Let ${\lambda }^{\ast }>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} Δ^{2}u=λ(1+u)^{p} & {in}\ \ \B, %0<u\leq 1 & {in}\ \ \B, u=\frac{\partial u}{\partial n} =0 & {on}\ \ \partial \B {array}. $$ has a solution, where $\B$ 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 $u^{\ast }(x)\le r^{-\frac{4}{p-1}}-1$ on the unit ball.

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