# Relation between the norm of Sobolev space $H^1$ and $L^p$ norm for non-increasing radial functions

I am interested to find $$\sup\|u\|_{p}^{p},$$ when $$u$$ are non-increasing radial functions on the unit ball $$B_1$$ of $$\mathbb{R}^{n}$$ such that $$\|u\|_{H1}^2 < r$$ for some $$r > 0$$. Since $$u$$ can be a constant function, we can prove that $$\sup\|u\|_{p}^{p}≥ r^{p\over 2}(Vol B_1)^{1-{p\over 2}}.$$ My question is: can we prove that $$\sup\|u\|_{p}^{p}> r^{p\over 2}(Vol B_1)^{1-{p\over 2}}\;???$$

Thank you for any help.