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 $${\displaystyle \underset{x\mathrm{\ne }y}{\sup }\frac{\mathrm{\mid }u(x)-u(y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1-\frac{n}{p}}}=C(p,n)\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)},}$$ where $p>n$ and $C(p,n)$ is the optimal constant in Morrey’s inequality. We prove that if $n\ge 2$ then any extremal has a power decay of order $\beta$ for any $$\beta <-\frac{1}{3}+\frac{2}{3(p-1)}+\sqrt{{(-\frac{1}{3}+\frac{2}{3(p-1)})}^2+\frac{1}{3}}\mathrm{.}$$