Set $p=2+4/d$. For $u_{a,\lambda}(x)=au(\lambda x)$, the <change of variables> $y=\lambda x$ gives
$$
\|\nabla u_{a,\lambda}\|_2^2
=|a|^2\lambda^{2-d}\|\nabla u\|_2^2,\qquad
\|u_{a,\lambda}\|_2^{4/d}
=|a|^{4/d}\lambda^{-2}\|u\|_2^{4/d},
$$
and
$$
\|u_{a,\lambda}\|_p^p
=|a|^{2+4/d}\lambda^{-d}\|u\|_p^p.
$$
The factors cancel, so the <Weinstein functional> satisfies $J(u_{a,\lambda})=J(u)$.

The <Gagliardo-Nirenberg interpolation inequality> gives
$$
\|u\|_p^p\leq C_d\|\nabla u\|_2^2\|u\|_2^{4/d}.
$$
Consequently $J(u)\geq C_d^{-1}$ for every nonzero $u\in H^1$, and therefore $I>0$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-154/2/1/solution