Scattering for the $2d$ NLS with inhomogeneous nonlinearities

Abstract

We prove large-data scattering in $H^1$ for inhomogeneous nonlinear Schr\"odinger equations in two space dimensions for all powers $p>0$. We assume the inhomogeneity is nonnegative and repulsive; we additionally require decay at infinity in the case $0<p\leq 2$. We use the method of concentration-compactness and contradiction. We preclude the existence of compact solutions using a Morawetz estimate in the style of Nakanishi.