Global dynamics for a class of inhomogeneous nonlinear Schrödinger equations with potential

Abstract

We consider a class of L2‐supercritical inhomogeneous nonlinear Schrödinger equations with potential in three dimensions. In the focusing case, using a recent method of Dodson and Murphy, we first study the energy scattering below the ground state for the equation with radially symmetric initial data. We then establish blow‐up criteria for non‐radial solutions to the equation. In the defocusing case, we also prove the energy scattering for the equation with radially symmetric initial data.

Scattering for the inhomogeneous nonlinear Schrödinger equation with a potential

“…It was proved for the INLS (1.5) in [3]. For the INLS with a potential, it was proved for radial solutions in [5]. Here, we prove the result for non-radial initial data the full subcritical range in three dimensions.…”

“…Let us recall some properties related to the ground state Q which is the unique positive radial decreasing solution to the elliptic equation( [3], [5]) 13) and (1.14), then the corresponding solution to the focusing problem (1.1) satisfies…”

“…where Q is ground state, i.e. the unique positive radially symmertric solution to the elliptic equation ( [3], [5])…”

“…When b = 0, V. D. Dinh [5] proved the global dynamics for a class L 2 -supercitical nonlinear inhomogeneous Schrödinger equation with a potential in three dimensions with radially symmetric initial data. Inspired by L.Campos and M. Cardoso [3], we will study the the well-posedness and scatterring for this problem with any initial data , which expends the result of V. D. Dinh [5].…”

Dynamics of the energy critical nonlinear Schrödinger equation with inverse square potential

“…(1.3) has also been studied by several authors in recent years. For example, Dinh [11] studied the well-posedness, scattering and blowup for (1.3) when d = 3, b > 0, λ = ±1 and V is a real-valued potential satisfying V ∈ K 0 ∩ L 3 2 and V − < 4π, where V − := min {V, 0} and K 0 is defined as the closure of bounded and compactly supported functions with respect to the Kato norm…”