Title: 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.