Energy-critical inhomogeneous nonlinear Schr\"odinger equation with two power-type nonlinearities

We consider the initial value problem for the inhomogeneous nonlinear Schr\"odinger equation with double nonlinearities (DINLS) equation* i ₜ u + u = ₁ |x|^-b₁|u|^p₁u + ₂|x|^-b₂|u|^4-2b₂{N-2}u, equation* where ₁, ₂ R, 3 N<6 and 0<b₁, b₂<\2, 6-N{2\}. In this paper, we establish global well-posedness results for certain parameter regimes and prove finite-time blow-up phenomena under specific conditions. Our analysis relies on stability theory, energy estimates, and virial identities adapted to the DINLS model.