Radial 3D Focusing Energy Critical INLS equations with defocusing perturbation: Ground states, Scattering, and Blow-up

Abstract: We investigate the following inhomogeneous nonlinear Schr\"odinger equation in the radial regime, featuring a focusing energy-critical nonlinearity and a defocusing perturbation: $$ i\partial_t u +\Delta u =|x|^{-a} |u|^{p-2} u - |x|^{-b} |u|^{4-2b}u \quad \mbox{in} \,\, \mathbb{R}_t \times \mathbb{R}_x^3, $$ where $0<a$, $b<2$ and $2+\frac{4-2a}{3}< p\leq 6-2a$. First, we establish the existence and nonexistence of ground states, along with their quantitative properties. Subsequently, we analyze the dichotomy between scattering and blow-up for solutions with energy below the ground-state energy threshold. An intriguing feature of this equation is the lack of scaling invariance, which arises from the competing effects of the inhomogeneous nonlinearities. Additionally, the presence of singular weights breaks translation invariance in the spatial variable, introducing further complexity to the analysis. To the best of our knowledge, this work represents the first comprehensive study of the inhomogeneous nonlinear Schr\"odinger equation with a leading-order focusing energy-critical inhomogeneous nonlinearity and a defocusing perturbation. Our results provide new insights into the interplay between these competing nonlinearities and their influence on the dynamics of solutions.