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

Question

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.