Profile decomposition modulo translations
= Profile decomposition modulo translations
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 $2<p<2^*$.