Semiclassical states for fractional Schrodinger equations with critical growth

In this paper, we consider the following fractional nonlinear Schrodinger equations epsilon(2s)(-Delta)(s)u + V(x)u = P (x)g(u) + Q(x)vertical bar u vertical bar(2s)*(-2) u, x is an element of R-N and prove the existence and concentration of positive solutions under suitable assumptions on the potentials V(x), P(x) and Q(x). We show that the semiclassical solutions ye with maximum points x(epsilon) concentrating at a special set S-P characterized by V (x), P (x) and Q(x). Moreover, for any sequence x(epsilon) -> x(0) is an element of S-P, v(epsilon)(x) := u(epsilon)(epsilon x+x(epsilon)) convergence strongly/ in H-s(R-N) to a ground state solution v of (-Delta)(s)v+V(X-0)v = P(x(0))g(v) + Q(x(0))vertical bar v vertical bar(2s*-2)v, x is an element of R-N. (C) 2016 Elsevier Ltd. All rights reserved.