Properties of the extremal solution for a fourth-order elliptic problem

Abstract

Let ${\lambda }^{\ast }>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} Δ^{2}u=\fracλ{(1-u)^{p}} & \{in}\ \ B, 0<u\leq 1 & \{in}\ \ B, u=\frac{\partial u}{\partial n} =0 & \{on}\ \ \partial B. {array} . $$ has a solution, where $B$ is the unit ball in $R^n$ centered at the origin, $p>1$ 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 and $1-C_0{r}^{\frac{4}{p+1}}\le u^{\ast }(x)\le 1-r^{\frac{4}{p+1}}$ on the unit ball, where $C_0:=({\lambda }^{\ast }/\bar{\lambda }{)}^{\frac{1}{p+1}}$ and $\bar{\lambda }:=\frac{8(p-1)}{(p+1{)}^2}[n-\frac{2(p-1)}{p+1}][n-\frac{4p}{p+1}]$. Our results actually complete part of the open problem which \cite{D} lef

18 pages 2figures