  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 $p>n$.
