Asymptotic flatness of Morrey extremals

Abstract

We study the limiting behavior as $|x|\rightarrow \infty$ of extremal functions $u$ for Morrey's inequality on $\mathbb{R}^n$. In particular, we compute the limit of $u(x)$ as $|x|\rightarrow \infty$ 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 $-\Delta_pu=c(\delta_{x_0}-\delta_{y_0})$ 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 $p>n$.