Global existence and asymptotic behavior of the Cauchy problem for fourth-order Schrödinger 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 iut+Δ2u+λ1|u|pu+λ2|u|8n−4u=0, where t∈Rt∈R, x∈Rnx∈Rn, 8n≤p<8n−4, λ1,λ2λ1,λ2 are nonzero real numbers.