By exploiting the prior information of azimuth-elevation (AE), a sparse recovery (SR)-based space-time adaptive processing (STAP) method called AESR-STAP is proposed. First, the supercomplete properties matrix in the SR is built based on flight configuration prior information such as elevation and azimuth to estimate the azimuth-elevation spectrum of the ground clutter in different range cells. Then, the range ambiguity STAP clutter can be eliminated according to the difference of the elevation of the clutter in different range cells. Finally, the AESR-STAP utilizes the spatial and temporal coupling relation of the ground clutter, namely, the nonlinear relation among the azimuth, the elevation, Doppler frequency, and spatial frequency, to estimate the clutter distribution in time and space dimensions and to calculate the clutter covariance matrix. Theoretical analysis and simulation experiments have shown that the proposed method can estimate the temporal and spatial distribution more accurately, eliminate the range ambiguity STAP clutter to improve the clutter suppression of the airborne radar, and effectively detect ground low-speed targets.