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We study quantitative aspects and concentration phenomena for ground statesof the following nonlocal Schr\"odinger equation $  (-\Delta)^s u+V(x)u= u^{2_s^*-1-\varepsilon} \ \  \text{in}\ \ \mathbb{R}^N, $ where $\varepsilon>0$, $s\in (0,1)$,$2^*_s:=\frac{2N}{N-2s}$, $N>4s$. We show that the ground state$u_{\varepsilon}$ blows up and precisely with the following rate$\|u_{\varepsilon}\|_{L^\infty(\mathbb{R}^N)}\sim\varepsilon^{-\frac{N-2s}{4s}}$, as $\epsilon\rightarrow 0^+$. We also localizethe concentration points and, in the case of radial potentials $V$, we provelocal uniqueness of sequences of ground states which exhibit a concentratingbehavior.

Blow-up rate and local uniqueness for fractional Schr\"odinger equations with nearly critical growth