Typical formulations of max-min weighted SIR problems involve either a total power constraint or individual power constraints. These formulations are unable to handle the complexities in multicell networks where each base station can be subject to its own sum power constraint. This paper considers the max-min weighted SIR problem subject to multiple weighted-sum power constraints, where the weights can represent relative power costs of serving different users. First, we derive the uplink-downlink duality principle by applying Lagrange duality to the single-constraint problem. Next, we apply nonlinear Perron-Frobenius theory to derive a closed-form solution for the multiple-constraint problem. Then, by exploiting the structure of the closed-form solution, we relate the multiple-constraint problem with its single-constraint subproblems and establish the dual uplink problem. Finally, we further apply nonlinear Perron-Frobenius theory to derive an algorithm which converges geometrically fast to the optimal solution.