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) \begin{equation*} i \partial_t u + \Delta u = \lambda_1 |x|^{-b_1}|u|^{p_1}u + \lambda_2|x|^{-b_2}|u|^{\frac{4-2b_2}{N-2}}u, \end{equation*} where $\lambda_1,\lambda_2\in \mathbb{R}$, $3\leq N<6$ 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.