To　deal　with　the　power　control　problem　in　cognitive　radio　(CR)　networks,　in　this　paper,　the　non-cooperative　power　game　is　studied,　in　which　each　CR　user　distributedly　and　asynchronously　updates　its　transmission　power.　Taking　into　account　the　different　quality　of　service　(QoS)　requirements,　with　the　objective　as　maximizing　the　average　utility　of　each　CR　user　under　the　interference　power　constraints　of　primary　users,　we　propose　a　QoS-aware　power　control　game　scheme　and　introduce　the　sigmoid　efficiency　function　related　with　CR　user＇s　signal-to-noise　ratio　threshold　and　the　price　function.　Besides,　extensive　simulation　results　are　presented　to　show　the　significant　performance　improvement　of　the　proposed　scheme　compared　to　the　existing　ones.