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

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrodinger equation (-Delta)(s)u + V (x)u = u(2* s -1-epsilon) in R-N, where epsilon > 0, s is an element of (0, 1), 2*(s) := 2N/N-2s and N > 4s, as we deal with finite energy solutions. We show that the ground state u blows u(epsilon) and precisely with the following rate parallel to u(epsilon)parallel to(L infinity (RN)) similar to epsilon-(N-2s/4s), as epsilon -> 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.