fa.functional analysis – relation between the norm of Sobolev space H1 and Lp norm for non-increasing radial function Answer

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fa.functional analysis – relation between the norm of Sobolev space H1 and Lp norm for non-increasing radial function

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

Thank you for any help.

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