Asymptotic Behavior of Positive Solutions of the Equation $\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in R^n and Positive Scalar Curvature

Abstract

We study asymptotic behavior of positive smooth solutions of the conformal scalar curvature equation in ${\bf R}^n$. We consider the case when the scalar curvature of the conformal metric is bounded between two positive numbers outside a compact set. It is shown that the solution has slow decay if the radial change is controlled. For a positive solution with slow decay, the corresponding conformal metric is found to be complete if and only if the total volume is infinite. We also determine the sign of the Pohozaev number in some situations and show that if the Pohozaev is equal to zero, then either the solution has fast decay, or the conformal metric corresponding to the solution is complete and the corresponding solution in ${\bf R} \times S^{n - 1}$ has a sequence of local maxima that approach the standard spherical solution.