 Compactness of a mass-critical minimizing sequence

Suppose $\Vert u_n{\Vert }_2=\Vert Q{\Vert }_2$, $\Vert \nabla u_n{\Vert }_2=\Vert \nabla Q{\Vert }_2$, and $E(u_n)\to 0$. The profile decomposition modulo translations and the sharp inequality force exactly one nonzero profile: splitting the mass would make the sharp inequality strict. Hence, after passage to a subsequence, some translations $u_n(\cdot +x_n)$ converge strongly in $H^1$, and therefore in $L^{2+4/d}$, to an optimizer.