Some Results on the Scattering Theory for a Schr\"{o}dinger Equation with Combined Power-Type Nonlinearities

TL;DR

This paper investigates scattering properties for a nonlinear Schrödinger equation with combined power nonlinearities, advancing understanding of open problems in the field and establishing scattering theory in a weighted Sobolev space.

Contribution

It introduces new techniques to analyze scattering for Schrödinger equations with combined nonlinearities, partly solving open problems posed by Tao, Visan, and Zhang.

Findings

Established scattering results for equations with combined power nonlinearities.

Extended scattering theory to the weighted Sobolev space  in .

Solved an open problem for pure power nonlinear Schrödinger equations in .

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.