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

Abstract: In this paper, we consider the Cauchy problem {align*} {{array}{ll}&i u_t+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=\phi(x)\in \Sigma, \quad x\in\mathbb{R}^N, {array}. {align*} where $N\geq 3$, $0<p_1<p_2\leq\frac{4}{N-2}$, $\lambda_1\in\mathbb{R}\setminus\{0\}$ and $\lambda_2\in\mathbb{R}$ are constants, $\Sigma=\{f\in H^1(\mathbb{R}^N); |x|f\in L^2(\mathbb{R}^N)\}$. 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 Schr\"{o}dinger equation with combined power-type nonlinearities, Communications in Partial Differential Equations, 32(2007), 1281--1343]. As a byproduct, we establish the scattering theory in $\Sigma$ for {align*} \{{array}{ll}&i u_t+\Delta u=\lambda|u|^pu, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=\phi(x), \quad x\in\mathbb{R}^N {array}. \={align*} with $\lambda\>0$ and $\frac{2}{N}<p<\alpha_0$ with $\alpha_0=\frac{2-N+\sqrt{N^2+12N+4}}{2N}$, which is also an open problem in this direction.