In this paper, the performance of an orthogonal frequency division multiplexing overlay cognitive radio network with subcarrier selection schemes is investigated. We propose three subcarrier selection techniques that reduce the level of interference at the primary base station based on collected channel state information from the different network nodes. Approximated outage probability expressions are also derived and verified by simulations for the different studied techniques. In addition, we propose and investigate a new approach for asynchronous narrowband interference (NBI) estimation and mitigation in cognitive radio networks. The proposed approach does not require prior knowledge of the NBI characteristics and allows the primary user to exploit the sparsity of the secondary users interference to recover it based on sparse signal recovery theory and approach the interference-free limit over practical ranges of NBI power levels.