\leqslant \Bigl\|  \frac{\| u(\cdotp -y) - u\|_{L^{p}}}{|y| ^s} \Bigr\|_{L^r(\R^d ; \frac{dy}{|y| ^d})} \, \leqslant C\,  \| u\|_{\dot{B}^{s}_{p,r}}.$$

Notice that $\dot{H}^s  \subset \dot{B}^{s}_{2,2}$ and both spaces coincide if ${\displaystyle}{s < \frac{3}{2}\cdotp}$

We recall an interpolation property in Besov spaces, which will be useful in the sequel.

*[ \[interpolation\] A constant $C$ exists which satisifes the following property. If $s_{1}$ and $s_{2}$ are real numbers such that $s_{1} < s_{2}$ and $\theta \in ]0,1[$, then we have for any $p\in [1,+\infty]$\
$$\| u\|_{\dot{B}^{\theta \,s_{1}  + (1-\theta)\, s_{2} }_{p,1}} \,\, \leqslant\,\,  C(s_{1},s_{2},\theta)\,  \| u\|^{\theta}_{\dot{B}^{s_{1}}_{p,\infty}}  \,\,  \| u\|^{1-\theta}_{\dot{B}^{s_{2}}_{p,\infty}}.$$ ]{}*

Application of profile theory to sup-critical solutions
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This section is devoted to the proof of Lemma \[general lemma for critical element \]. The statement given below is actually a bit stronger and clearly entails Lemma \[general lemma for critical element \]. We shall prove the following proposition.

*[ \[proposition critical element \] Let $(v_{0,n})_{n \in \N}$ be a bounded sequence in $\dot{H}^s$ such that $$\tau^{*}(v_{0,n}) = 1 \quad \hbox{ and} \quad \hbox{ for any} \,\,\, \tau < 1, \quad (1-\tau)\, \| NS(v_{0,n})(\tau,) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant\,\, M^{\sigma_{s}}_{c} + \varepsilon_{n},$$ where $\varepsilon_{n}$ is a generic sequence which tends to $0$ when $n$ goes to $+\infty$.\
Then, up 