The Blow-up Behavior Of Semi-linear Elliptic Equation With Neumann Boundary

In this paper, we mainly study the blow-up behavior of the Elliptic Partial Differential EquationThis problem is originated from conformal mapping theory. It is of great use to find out whether there is a conformal mapping from 2-dimension Riemann Surface to part of ball. First, by simplifying general equation ?Δu + R0 = eu and by conformal mapping from ?M 0 to a number of upper half circle, we can focus on the problem with u t = f ( x ) on {t = 0}∩?B R+ and u = 0 on ?B R +∩BR+. Second, establish two crucial inequalities, namely L∞Estimate and Harnack Inequality. At last, we get a blow-up theory. If { are solutions of equations with andThere exists a subsequence satisfying alternative:1) unk is bounded in ; L∞( M0)2) uniformly on any compact subsets of u nk→?∞M 0;3) is finite,nonempty and uniformly on any compact subset of .