Suppose $\|u_n\|_2=\|Q\|_2$, $\|\nabla u_n\|_2=\|\nabla Q\|_2$, and $E(u_n)\to0$. The <profile decomposition modulo translations> and the sharp inequality force exactly one nonzero profile: splitting the mass would make the sharp inequality strict. Hence some translations $u_n(\cdot+x_n)$ converge strongly in $H^1$, and therefore in $L^{2+4/d}$, to an optimizer.

Compactness of a mass-critical minimizing sequence

The <profile decomposition modulo translations> says that every bounded sequence $(w_n)$ in $H^1(\mathbb R^d)$ has, after passage to a subsequence,
$$
w_n=\sum_{j=1}^J\phi^j(\,\cdot-x_n^j)+r_n^J,
$$
where the translation parameters are asymptotically orthogonal,
$$
|x_n^j-x_n^k|\longrightarrow\infty\quad(j\ne k),
$$
the squared $L^2$ and gradient norms decouple,
$$
\|w_n\|_2^2=\sum_{j=1}^J\|\phi^j\|_2^2+\|r_n^J\|_2^2+o_n(1),
$$
$$
\|\nabla w_n\|_2^2
=\sum_{j=1}^J\|\nabla\phi^j\|_2^2+\|\nabla r_n^J\|_2^2+o_n(1),
$$
and the remainder vanishes in every subcritical norm:
$$
\lim_{J\to\infty}\limsup_{n\to\infty}
\|r_n^J\|_{L^p}=0
\qquad(2<p<2^*).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 3 / 3 / 4 / Solution

Every bounded sequence $(u_n)$ in $H^1(\mathbb R^d)$ has, after passage to a subsequence, a decomposition
$$
u_n=\sum_{j=1}^J\phi^j(\,\cdot-x_n^j)+r_n^J,
$$
where $|x_n^j-x_n^k|\to\infty$ for $j\ne k$, the $L^2$ and gradient norms decouple asymptotically, and
$$
\lim_{J\to\infty}\limsup_{n\to\infty}
\|r_n^J\|_{L^p}=0
$$
for every $2<p<2^*$.

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 Incoming links: Profile decomposition modulo translations