This paper studies downlink coverage maximization for cellular networks in which base station (BS) locations are modeled using a spatial Poisson point process, considering three different coverage models, and under constraints on transmit power, BS density and transmit power density. Firstly, the coverage optimization problem is solved analytically for the first coverage model that focuses on noise-limited communication by ignoring interference and random fading effects. This model provides useful insights into the significance of bounded path loss models to obtain meaningful solutions for this problem. The other two coverage models are based on the users' received signal-to- interference-plus-noise-ratio (SINR) from their associated BSs. For these models, it is shown that the coverage optimization problem can be reduced to a constrained single dimensional optimization problem without any loss of optimality. The related solutions can be obtained with limited computational complexity by resorting to a numerical search over a compact subset of candidate values. Bounds on the optimum BS density are also provided to further truncate the search space. All results are derived for general bounded path loss models. Specific applications are also illustrated to provide further design insights and to highlight the importance of using bounded path loss models for coverage analysis.