Stable solutions for the bilaplacian with exponential nonlinearity

Abstract

Let ${\lambda }^{\ast }>0$ denote the largest possible value of $\lambda$ such that \begin{align*} \left\{\begin{aligned} Δ^2 u & = \la e^u && \text{in $B$} u &= \pd{u}{n} = 0 && \text{on $ \pa B $} \end{aligned} \right. \end{align*} has a solution, where $B$ is the unit ball in ${\mathbb{R}}^N$ and $n$ is the exterior unit normal vector. We show that for $\lambda ={\lambda }^{\ast }$ this problem possesses a unique {\em weak} solution $u^{\ast }$. We prove that $u^{\ast }$ is smooth if $N\le 12$ and singular when $N\ge 13$, in which case $u^{\ast }(r)=-4\log r+\log (8(N-2)(N-4)/{\lambda }^{\ast })+o(1)$ as $r\to 0$. We also consider the problem with general constant Dirichlet boundary conditions.