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In cognitive radio networks, power control is necessary to not only decrease the interference among the secondary users (SUs), but also avoid negative impact to the primary users (PUs). Prevalent research works on power control are mainly focus on maximizing SINR as the QoS requirement of SUs under the interference power constraint for PUs. We note that besides achieving a high SINR to guarantee reliable data transmissions, SUs also require to support heterogenous services with different transmission rates. In order to provide flexible transmission rates to each SU, efficient use of networks radio resource requires transmission rate control in addition to transmit power control. In this paper, we consider the problem of joint power and rate control for SUs in cognitive radio network by using non-cooperative game theory. We study how to jointly allocate optimal transmit power and transmission rate given certain QoS requirement of SUs. We analysis of existence, uniqueness and Pareto efficiency of Nash equilibrium for our game. The performance of our proposed joint power and rate control algorithm is investigated by numeral results.