I try to prove this lemma:
Let $\mathrm{H}_{\text{comp}}^1(\Bbb R^{\mathrm{N}})$ be the subspace of $\mathrm{H}^1(\Bbb R^{\mathrm{N}})$ of functions with compact support. For each $u\in\mathrm{H}_{\text{comp}}^1(\Bbb R^{\mathrm{N}})$ and $u\ne 0$, let
$$\beta(u)=\frac{\int_{\Bbb R_\mathrm{N}}|\nabla u(x)|^2x\,dx}{\int_{\Bbb R_\mathrm{N}}|\nabla u(x)|^2\,dx}\;.$$
From known results about compactness (see [$8$] and [$5$], theorem I.$6$) it is easy to prove the following lemma:
Lemma $\mathbf{1.1.}$ – For each $\rho>0$ there is an $\varepsilon\in(0,1)$ such that
$$\frac{\|u\|^2}{|u|_{2^*}^{2^*}}\le\mathrm{S}+\varepsilon\quad\implies\quad\beta\left(\frac{u}{|u|_{2^*}}\right)\in\Omega_\rho^+\;.$$
Where [$8$] is here, and [$5$] is here.
Can you tell me how we can prove this?
Thank you.