This paper studies unmanned aerial vehicle (UAV) aided wireless communication systems where a UAV serves uplink communications of multiple ground nodes by flying the area of the interest. We aim to maximize the minimum average rate of the UAV by jointly optimizing the UAV trajectory and the ground nodes' uplink transmit power. However, this problem is shown to be non-convex in general, and thus existing convex optimization techniques and algorithms cannot be directly applied. By employing the successive convex approximation (SCA) techniques, we present an efficient algorithm which is guaranteed to converge to at least a local optimal point for the non-convex problems. To this end, proper convex approximations are derived for the non-convex constraints. Numerical results demonstrate the proposed algorithm performs better than baseline scheme.