Moving planes, moving spheres, and a priori estimates

In this paper, we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi-linear elliptic equations. Using the ‘method of moving planes’, we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of Lin. Applying ‘the method of moving spheres’, we give a different proof for a well-known sup+inf inequality established by Brezis, Li and Shafrir.