On the semirelativistic Hartree-type equation

We study the global Cauchy problem and scattering problem for the semirelativistic Hartree-type 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 nonexistence 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.