Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions

Abstract

In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [10], the first and the second authors consider one- and two-dimensional cases and gave a sufficient condition on the nonlinearity for that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converges to a given asymptotic profile in a faster rate than in the lower dimensional cases. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in [10]. Moreover, we present a candidate of the second asymptotic profile to the solution.

23 pages

Cited by in corpus (7)

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• Threshold for the existence of scattering states for nonlinear Schrödinger equations without gauge invariance
• Dynamics near the origin of the long range scattering for the one-dimensional Schrodinger equation
• Modified scattering for nonlinear Schrödinger equations with long-range potentials