This paper considers a fading cognitive multiple access channel (CMAC), where multiple secondary users (SUs), who share the spectrum with a primary user (PU), transmit to a cognitive base station (CBS). A power station is assumed to harvest energy from the nature and then provide power to the SUs. We investigate the power allocation problems for such a CMAC to maximize the SU sum rate under the interference power constraint, the sum transmit power constraint and the peak transmit power constraint of each individual SU. In particular, two scenarios are considered: with successive interference cancellation (SIC) and without SIC. For the first scenario, the optimal power allocation algorithm is derived. For the second scenario, a heuristic algorithm is proposed. We show that the proposed algorithm with SIC outperforms the algorithm without SIC in terms of the SU sum rate, while the algorithm without SIC outperforms the algorithm with SIC in terms of the number of admitted SUs for a high sum transmit power limit and a low peak transmit power limit of each individual SU.