a sequence of scales and cores $(\lambda_{n,j},x_{n,j})_{n,j \in \N}$, such that, up to an extraction $$\forall J \geqslant 0,\,\, u_{0,n}(x) = \sum_{j=0}^{J} \Lambda^{\frac{3}{p}}_{\lambda_{n,j},x_{n,j}}\varphi^{j}(x) + \psi_{n}^{J}(x) \quad \hbox{with}\quad \lim_{J \to +\infty}\limsup_{n \to+\infty}\|\psi_{n}^{J}\|_{L^{p}} =0, \quad \hbox{and} \quad p =\frac{6}{3-2s}\cdotp$$ Where, $(\lambda_{n,j},x_{n,j})_{n \in \N,j \in \N^*}$ are sequences of $(\R_{+}^* \times \R^3)^{\N}$ with the following orthogonality property: for every integers $(j,k)$ such that $j \neq k$, we have $$\hbox{either}\lim_{n \to +\infty}\Bigl(\frac{\lambda_{n,j}}{\lambda_{n,k}} + \frac{\lambda_{n,k}}{\lambda_{n,j}}\Bigr) = +\infty  \quad\hbox {or} \quad  \lambda_{n,j} = \lambda_{n,k} \quad\hbox {and} \quad \lim_{n \to +\infty}\frac{|x_{n,j} - x_{n,k}|}{\lambda_{n,j}} = +\infty.$$ Moreover, for any $J \in \N$, we have the following orthogonality property $$\label{ortho de la norme}
\| u_{0,n} \|^2_{\dot{H}^s} = \sum_{j=0}^{J} \| \varphi^{j} \|^2_{\dot{H}^s} + \| \psi_{n}^J \|^2_{\dot{H}^s} + \circ(1), \quad\hbox {when} \quad n \to +\infty.$$ ]{}*

0.3cm Let us recall a structure Lemma, based on the crucial profils theorem of P. Gérard (see [@PG]). Let $(u_{0,n})$ be a bounded sequence in the Sobolev space $\dot{H}^s$, which profile decomposition is given by $$u_{0,n}(x) =  \sum_{j \in J}  \Lambda^{\frac{3}{p}}_{\lambda_{n,j},x_{n,j}} \varphi^{j}(x) + \psi_{n}^{J}(x),$$ with the appropriate properties on the error term  $\psi_{n}^{J}$. By vertue of orthogonality of scales and cores given by Theorem \[theo profiles\], we sort profiles according to their scales $$\label{decomposition 1}
 