Decay and scattering of solutions to nonlinear Schrödinger equations with regular potentials for nonlinearities of sharp growth

Abstract

In this paper, we prove the decay and scattering in the energy space for nonlinear Schrödinger equations with regular potentials in ${\mathbb{R}}^d$ namely, $i{\partial }_{t}u+\Delta u-V(x)u+\lambda |u{|}^{p-1}u=0$. We will prove decay estimate and scattering of the solution in the small data case when $1+\frac{2}{d}<p\le 1+\frac{4}{d-2}$, $d\ge 3$. The index $1+\frac{2}{d}$ is sharp for scattering concerning the result of W. Strauss [21].

arXiv comment22 pages