Concentrating standing waves for the fractional Schrodinger equation with critical nonlinearities

We study the following nonlocal Schrodinger equations: epsilon(2s)(-Delta)(s)u+V(x)u = W(x)f(u), epsilon(2s)(-Delta)(s)u+V(x)u = W(x)f(u) + u(2s*-1)) for u is an element of H-s(R-N), where f (u) is superlinear and subcritical, 2s*=2N/N-2s if N>2s. V(x) and W(x) are sufficiently smooth potential with inf V(x) > 0, inf W(x) > 0, and epsilon > 0 is a small number. Under proper assumptions, we explore the existence, concentration phenomenon, convergence, and decay estimate of semiclassical solutions of (I) and (II), respectively. Compared with some existing issues, the most interesting results obtained here are therefore: the concentration phenomenon depends on competing potential functions; the nonlocal critical problem (II) is considered; unlike the classical case s = 1, the decay estimate of solution to (I) or (II) is of polynomial instead of exponential form, due to the nonlocal effect.