Asymptotic flatness of Morrey extremals

Abstract

We study the limiting behavior as $|x|\to \infty$ of extremal functions $u$ for Morrey's inequality on ${\mathbb{R}}^n$. In particular, we compute the limit of $u(x)$ as $|x|\to \infty$ and show $|x||Du(x)|$ tends to $0$. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form $-{\Delta }_{p}u=c({\delta }_{x_0}-{\delta }_{y_0})$ for some $c\in \mathbb{R}$ and distinct $x_0,y_0\in {\mathbb{R}}^n$. More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded $p$-harmonic functions on exterior domains of ${\mathbb{R}}^n$ for $p>n$.