On the semi-relativistic Hartree type equation

Abstract. We study the global Cauchy problem and scattering problem for the semi-relativistic equation in Rn, n ≥ 1 with nonlocal nonlinearity F (u) = λ(|x|−γ ∗ |u|2)u, 0 < γ < n. We prove the existence and uniqueness of global solutions for 0 < γ < 2n n+1, n ≥ 2 or γ> 2, n ≥ 3 and the non-existence of asymptotically free solutions for 0 < γ ≤ 1, n ≥ 3. We also specify asymptotic behavior of solutions as the mass tends to zero and infinity. 1