A Virial-Morawetz approach to scattering for the non-radial inhomogeneous NLS

Abstract

Consider the focusing inhomogeneous nonlinear Schrödinger equation in H 1 ( R N ) H^1(\mathbb {R}^N) , i u t + Δ u + | x | − b | u | p − 1 u = 0 , \begin{equation} iu_t + \Delta u + |x|^{-b}|u|^{p-1}u=0, \end{equation} when b > 0 b > 0 and N ≥ 3 N \geq 3 in the intercritical case 0 > s c > 1 0 > s_c >1 . In previous works, the second author, as well as Farah, Guzmán and Murphy, applied the concentration-compactness approach to prove scattering below the mass-energy threshold for radial and non-radial data. Recently, the first author adapted the Dodson-Murphy approach for radial data, followed by Murphy, who proved scattering for non-radial solutions in the 3d cubic case, for b > 1 / 2 b>1/2 . This work generalizes the recent result of Murphy, allowing a broader range of values for the parameters p p and b b , as well as allowing any dimension N ≥ 3 N \geq 3 . It also gives a simpler proof for scattering nonradial, avoiding the Kenig-Merle road map. We exploit the decay of the nonlinearity, which, together with Virial-Morawetz-type estimates, allows us to drop the radial assumption.