The nonlinear schrödinger equation with combined power-type nonlinearities

Abstract

We undertake a comprehensive study of the nonlinear Schrdinger equation [image omitted] where u(t, x) is a complex-valued function in spacetime [image omitted], 1 and 2 are nonzero real constants, and [image omitted]. We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H1(n) and in the pseudoconformal space :={fH1(n); xfL2(n)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the [image omitted]-critical, respectively [image omitted]-critical NLS, that is, 1, 20 and [image omitted], [image omitted]. The results at the endpoint [image omitted] are conditional on a conjectured global existence and spacetime estimate for the [image omitted]-critical nonlinear Schrdinger equation, which has been verified in dimensions n2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint). As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in [image omitted] for solutions to the nonlinear Schrdinger equation [image omitted] with [image omitted], which was first obtained by Ginibre and Velo (1985).