Concentrating solutions for a class of nonlinear fractional Schrödinger equations in RN

We deal with the existence of positive solutions for the following fractional Schrödinger equation: ε2s(−Δ)su + V (x)u = f(u) in RN , where ε > 0 is a parameter, s ∈ (0, 1), N ≥ 2, (−Δ)s is the fractional Laplacian operator, and V : RN → R is a positive continuous function. Under the assumptions that the nonlinearity f is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of V as ε tends to zero.