Power consumption is a major limitation in the downlink of multibeam satellite systems, since it has a significant impact on the mass and lifetime of the satellite. In this context, we study a new energy-aware power allocation problem that aims to jointly minimize the unmet system capacity (USC) and total radiated power by means of multi-objective optimization. First, we transform the original nonconvex-nondifferentiable problem into an equivalent nonconvex-differentiable form by introducing auxiliary variables. Subsequently, we design a successive convex approximation (SCA) algorithm in order to attain a stationary point with reasonable complexity. Due to its fast convergence, this algorithm is suitable for dynamic resource allocation in emerging on-board processing technologies. In addition, we formally prove a new result about the complexity of the SCA method, in the general case, that complements the existing literature where the complexity of this method is only numerically analyzed.