The definition of the <infimum> $I$ gives
$$
\|u\|_p^p
\leq I^{-1}\|\nabla u\|_2^2\|u\|_2^{4/d}.
$$
Substituting the value of $I$ from part 4 into the energy yields
$$
\begin{aligned}
E(u)
&=\frac12\|\nabla u\|_2^2-\frac1p\|u\|_p^p\\
&\geq\frac12\|\nabla u\|_2^2
\left[1-\left(\frac{\|u\|_2}{\|Q\|_2}\right)^{4/d}\right].
\end{aligned}
$$
This is <coercivity below the NLS ground-state mass>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 2 / 2 / 5 / Solution

The sharp inequality implies
$$
E(u)\geq\frac12\|\nabla u\|_2^2
\left[1-\left(\frac{\|u\|_2}{\|Q\|_2}\right)^{4/d}\right].
$$
Thus energy controls the gradient whenever the mass is strictly below the mass of the <NLS ground state>.

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