# Long range scattering for nonlinear Schr\"odinger equations with critical homogeneous nonlinearity in three space dimensions

## TL;DR

This paper investigates the long-range scattering behavior of solutions to three-dimensional nonlinear Schrödinger equations with critical homogeneous nonlinearities, establishing faster convergence rates and proposing a second asymptotic profile.

## Contribution

It extends previous two-dimensional results to three dimensions, employing new estimates and operators to analyze the asymptotic behavior of solutions.

## Key findings

- Solutions converge to a prescribed asymptotic profile faster than in lower dimensions.
- A candidate for the second asymptotic profile is proposed.
- The use of end-point Strichartz estimates and a time-dependent regularizing operator is crucial.

## Abstract

In this paper, we consider the final state problem for the nonlinear Schr\"odinger 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.