Estimates of the conformal scalar curvature equation via the method of moving planes

In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal equation $(0.1)\hfill\left\{\matrix{\Delta u+K(x)u^{{{n+2}\over{n-2}}} + g(u) = 0\hfill & \hbox{in } \Omega\setminus Z\cr u(x)> 0 \hbox{ and } u \in C^2\hfill&\hbox{in }\Omega\setminus Z\hfill\cr}\right.,$ where K ( x ) is a positive continuous function, Z is a compact subset of $\bar\Omega,$ , and g satisfies that $g(t)t^{{{n+2}\over{n-2}}}$ is nonincreasing for t > 0. Assuming that the order of flatness at critical points of K on Z is no less than ${{n-2}\over{2}}$ , we prove that, through the application of the method of moving planes, the inequality $(0.2)\hfill u(x) \leq cd(x, Z)^{-{{n-2}\over{2}}}$ holds for any solution of (0.1) with Cap( Z ) = 0. By the same method, we also derive a Harnack‐type inequality for smooth positive solutions. Let u satisfy $(0.3)\hfill\left\{\matrix{\Delta u+K(x)u^{{{n+2}\over{n-2}}} + g(u) = 0\hfill & \hbox{in } B_{3R}(0)\cr u(x)> 0 \hbox{ and } u \in C^2\hfill&\hbox{in }B_{3R}(0)\hfill\cr}\right.,$ Assume that the order of flatness at critical points of K is no less than n ‐ 2; then the inequality $(0.4)\hfill\mathop{\max\limits_{B_R}\,} u\cdot \mathop{\min\limits_{B_{2R}}\, } \leq {{C}\over{R^{n-2}}}$ holds for R ≤ 1. We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4). © 1997 John Wiley & Sons, Inc.