Profile decomposition modulo translations

Every bounded sequence $(u_n)$ in $H^1({\mathbb{R}}^d)$ has, after passage to a subsequence, a decompositionwhere $|x_n^j-x_n^k|\to \infty$ for $j\ne k$, the $L^2$ and gradient norms decouple asymptotically, andfor every $2<p<2^{\ast }$, where $2^{\ast }=2d/(d-2)$ for $d\ge 3$ and $2^{\ast }=\infty$ for $d=1,2$.