In this paper, we consider a sequence of multibubble solutions u k of the equationwhere h is a C 2,β positive function in a compact Riemann surface M, and ρ k is a constant satisfying lim k→+∞ ρ k = 8mπ for some positive integer m ≥ 1. We prove among other things that

We consider the following mean field equations:where M is a compact Riemann surface with volume 1, h is a positive continuous function on M, ρ is a real number, andwhere is a bounded smooth domain, h is a C 1 positive function on , and ρ ∈ R. Based on our previous analytic work [14], we prove, among other things, that the degree-counting formula for (0.1) is given by m−χ(M) m for ρ ∈ (8mπ, 8(m + 1)π ).

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.

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in R 3 with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in [11,1] that they could only blow up on the axis of symmetry. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound |v(x, t)| ≤ C * (r 2 − t) −1/2 for −T 0 ≤ t < 0 and 0 < C * < ∞ allowed to be large, we then prove that v is regular at time zero.