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Some Results on the Scattering Theory for a Schrödinger Equation with Combined Power-Type Nonlinearities

In this paper, we consider the Cauchy problem {align*} \{{array}{ll}&i u_t+Δu=λ_1|u|^{p_1}u+λ_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=ϕ(x)\in Σ, \quad x\in\mathbb{R}^N, {array}. {align*} where $N\geq 3$, $0<p_1<p_2\leq\frac{4}{N-2}$, $λ_1\in\mathbb{R}\setminus\{0\}$ and $λ_2\in\mathbb{R}$ are constants, $Σ=\{f\in H^1(\mathbb{R}^N); |x|f\in L^2(\mathbb{R}^N)\}$. Using the strategy in \cite{Cazenave2, Cazenave3} and taking some elementary techniques which differ from the pseudoconformal conservation law, we obtain some scattering properties, which partly solve the open problems of Terence Tao, Monica Visan and Xiaoyi Zhang[The nonlinear Schrödinger equation with combined power-type nonlinearities, Communications in Partial Differential Equations, 32(2007), 1281--1343]. As a byproduct, we establish the scattering theory in $Σ$ for {align*} \{{array}{ll}&i u_t+Δu=λ|u|^pu, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=ϕ(x), \quad x\in\mathbb{R}^N {array}. \={align*} with $λ>0$ and $\frac{2}{N}<p<α_0$ with $α_0=\frac{2-N+\sqrt{N^2+12N+4}}{2N}$, which is also an open problem in this direction.