Scattering for the Radial Schrödinger Equation with Combined Power-type and Choquard-type Nonlinearities

Abstract

In this paper, we show the scattering of the radial solution for the nonlinear Schrödinger equation with combined power-type and Choquard-type nonlinearities

$$\rm{i}u_{t}+\Delta u=\lambda_{1}\vert u\vert^{p_{1}-1}u+\lambda_{2}(I_{\alpha}\ast\vert u\vert^{p_{2}})\vert u\vert^{p_{2}-2}u.$$

in the energy space H¹(ℝ^N) for λ₁λ₂ = −1. We establish a scattering criterion for radial solution together with Morawetz estimate which implies the scattering theory. Results show that the defocusing perturbation terms does not determine the scattering solution in energy space.