Energy-critical inhomogeneous nonlinear Schrödinger equation with two power-type nonlinearities

We consider the initial value problem for the inhomogeneous nonlinear Schrödinger equation with double nonlinearities (DINLS) \begin{equation*} i \partial_t u + Δu = λ_1 |x|^{-b_1}|u|^{p_1}u + λ_2|x|^{-b_2}|u|^{\frac{4-2b_2}{N-2}}u, \end{equation*} where ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, $3\le N<6$ and $0<b_1,b_2<\min \{2,\frac{6-N}{2}\}$. 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.