Global existence and asymptotic behavior of the Cauchy problem for fourth-order Schrِdinger equations with combined power-type nonlinearities

We study the global existence of solutions in Sobolev spaces and the asymptotic behavior for the Cauchy problem of the following fourth-order Schrödinger equation with combined power-type nonlinearities i u t + Δ 2 u + λ 1 | u | p u + λ 2 | u | 8 n − 4 u = 0 , where t ∈ R , x ∈ R n , 8 n ≤ p < 8 n − 4 , λ 1 , λ 2 are nonzero real numbers.