Threshold for the existence of scattering states for inhomogeneous nonlinear Schr\"odinger equations without gauge invariance
We consider the asymptotic behavior of solutions to inhomogeneous nonlinear Schr\"odinger equations with non-gauge-invariant nonlinearities. The spatial coefficient in the nonlinear term may have different orders of singularity at the origin and decay at infinity. Under a coercivity condition involving the coefficient and the nonlinearity, we show that no scattering states exist for the equation below a Strauss-type threshold determined by the decay at infinity. Our class includes nonlinearities with a dominant non-oscillatory component. We also prove a complementary small-data scattering result above this threshold, using non-admissible Strichartz estimates in Lorentz spaces adapted to the singular coefficient.
math.AP↗