Uniform decay estimates for solutions of the Yamabe equation

We study positive solutions u of the Yamabe equation c_m Du-s( x) u+k( x) u^fracm+2m-2=0{c_{m} Delta u-sleft( xright) u+kleft( xright) u^{frac{m+2}{m-2}}=0}, when k(x) > 0, on manifolds supporting a Sobolev inequality. In particular we get uniform decay estimates at infinity for u which depend on the behaviour at infinity of k, s and the L ^Γ-norm of u, for some G 3 tfrac2mm-2{Gammageqtfrac{2m}{m-2}}. The required integral control, in turn, is implied by further geometric conditions. Finally we give an application to conformal immersions into the sphere.