Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 2 / 1 / Solution

Set $p=2+4/d$. For $u_{a,\lambda }(x)=au(\lambda x)$, the change of variables $y=\lambda x$ givesandThe factors cancel, so the Weinstein functional satisfies $J(u_{a,\lambda })=J(u)$.

The Gagliardo-Nirenberg interpolation inequality givesConsequently $J(u)\ge C_d^{-1}$ for every nonzero $u\in H^1$, and therefore $I>0$.