Ground states for fractional Schrödinger equations with critical growth

In this paper we study the existence and multiplicity of solutions for the critical fractional Schrödinger equation ε^2α(−Δ)^αu+V(x)u=|u|^2_α∗−2u+λf(u),  x∈R^N, where ε and λ are positive parameters, 0 < α < 1, (−Δ)^α denotes the fractional Laplacian of order α, N > 2α, 2_α^∗=((2N)/(N−2α)) is the fractional critical exponent; V is a positive continuous potential satisfying some conditions and f is a continuous subcritical nonlinear term. We prove that the equation has a nonnegative ground state solution and investigate the relation between the number of solutions and the topology of the set where V attains its minimum, for all sufficiently large λ and small ε. (paper)