On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup

Abstract

This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption.

Keywords

• schrödinger problem
• energy-critical
• nonlinear equations
• fixed point method
• global existence
• scattering
• blowup