This study investigates the capacity of multiple relay channels in different fading and shadowing environments under spectrum-sharing constraints. The authors consider that a secondary user (SU) is allowed to share the spectrum band with a primary user (PU) provided that the SU's transmit power remains below an interference power threshold set by the PU. Considering a scenario where the SU's transmitter and receiver cannot communicate directly, a relay node, chosen among a set of K terminals, helps transmitting data from the SU's transmitter to the destination. The SU's transmitter and chosen relay node adapt their corresponding transmission parameters so as to satisfy the interference-power constraint at the PU's receiver. The authors derive closed-form expressions for the ergodic capacity of the SU's channel in Rayleigh fading, Nakagami-m fading and lognormal shadowing environments. They further obtain the outage capacity assuming the aforementioned environments and the above-mentioned spectrum-sharing limitations. Numerical results are provided to reinforce our theoretical derivations.