Asymptotic Behavior of Ground States and Local Uniqueness for Fractional Schrödinger Equations with Nearly Critical Growth.

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation (− Δ) s u + V (x) u = u 2 s ∗ − 1 − 휖 in ℝ N , where 휖 > 0, s ∈ (0,1), 2 s ∗ : = 2 N N − 2 s and N > 4s, as we deal with finite energy solutions. We show that the ground state u_휖 blows up and precisely with the following rate ∥ u 휖 ∥ L ∞ (ℝ N) ∼ 휖 − N − 2 s 4 s , as 휖 → 0 + . We also localize the concentration points and, in the case of radial potentials V, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior. [ABSTRACT FROM AUTHOR]