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 + Δ u = λ_1 |x|-b_1|u|p_1u + λ_2|x|-b_2|u|4-2b_2/N-2u, {equation*} where λ₁,λ₂∈ R, 3≤ N<6 and 0<b₁,b₂<min\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.