Decay of extremals of Morrey’s inequality

We study the decay (at infinity) of extremals of Morrey’s inequality in $\mathbb{R}^n$. These are functions satisfying\[\underset{x \neq y}{\sup} \frac{\lvert u(x)-u(y) \rvert}{{\lvert x-y \rvert}^{1-\frac{n}{p}}} = C(p,n) {\lVert \nabla u (\mathbb{R}^n \rVert}_{L^p (\mathbb{R}^n)} \; \textrm{,}\]where $p \gt n$ and $C(p, n)$ is the optimal constant in Morrey’s inequality. We prove that if $n \geq 2$ then any extremal has a power decay of order $\beta$ for any\[\beta \lt - \frac{1}{3} + \frac{2}{3(p-1)} + \sqrt{\left( -\frac{1}{3} + \frac{2}{3(p-1)} \right)^2 + \frac{1}{3}} \; \textrm{.}\]