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In this paper, we study the optimal power allocation strategies to achieve the delay-limited capacity and outage capacity for fading channels in cognitive radio networks considering interference power constraint. The optimal power allocation to achieve delay-limited capacity is shown to be similar to channel inversion, and the derived delay-limited capacity is related to the channel statistics. Moreover, the optimal power allocation strategy to achieve the outage capacity is derived, and the corresponding minimum outage probability is evaluated for each fading scenario under the peak interference power constraint and the average interference power constraint respectively. Finally, the simulation results are presented to validate the proposed power allocation methods.