Uniform decay estimates for solutions of the Yamabe equation

Abstract

We study positive solutions u of the Yamabe equation ${c_{m} \Delta u-s\left( x\right) u+k\left( x\right) u^{\frac{m+2}{m-2}}=0}$, 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 ${\Gamma\geq\tfrac{2m}{m-2}}$. The required integral control, in turn, is implied by further geometric conditions. Finally we give an application to conformal immersions into the sphere.