Let $$ E(u)=\frac{1}{2}\int_{\mathbb R^n} |\nabla u|^2-\frac{1}{p+1}|u|^{p+1} dx $$ where $p$ is a positive constant. Define $$ I_\mu=\inf\{E(u): u\in H^2(\mathbb R^n), ||u||_{L^2}^2=\mu\}. $$ Assume $\{u_n\}$ is a minimizing sequence of $I_\mu$, i.e. $$ u_n\in H^2(\mathbb R^n),~~ ||u||_{L^2}^2=\mu,~~ E(u_n)\rightarrow I_\mu. $$ Why there are $\{y_n\}\subset \mathbb R^n$ such that $\{u_n(\cdot+y_n)\}$ is relatively compact in $H^2(\mathbb R^n)$ ?
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This question is a doubt when I read the 552th page of Cazenave, T.; Lions, Pierre-Louis, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85, 549-561 (1982). ZBL0513.35007.