We address the problem of resource allocation on the downlink of an OFDMA single-cell system under fairness constraints with limited channel state information (CSI). Target QoS corresponds to a minimum user data rate, a target bit-error rate and a maximum BER-outage probability. The channel model includes path-loss, shadowing, and fading. The only available CSI is the channel average gain of each user. This partial CSI defines a shadowed path-loss that yields a modified user distribution. Resource allocation is based on the shadowed user distribution that we characterize analytically. Thus, under the target QoS, we provide the optimal resource allocation that maximizes the user rate. Compared to full-CSI-based allocation schemes, our solution offers a significant complexity and feedback reduction. Finally, the performance of our method is compared to other existing methods and the robustness of its outage performance to CSI errors is shown.