Stable solutions for the bilaplacian with exponential nonlinearity

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to a biharmonic equation with exponential nonlinearity in a ball, revealing a critical dimension at N=12 where solutions transition from smooth to singular.

Contribution

It establishes the existence of a unique weak solution at the critical parameter and characterizes the regularity and singularity of solutions depending on the dimension.

Findings

Unique weak solution at λ=λ*

Smooth solutions for N≤12

Singular solutions for N≥13 with explicit asymptotics

Abstract

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