Degree counting and Shadow system for Toda system of rank two: One bubbling

Abstract

We initiate the program for computing the Leray-Schauder topological degree for Toda systems of rank two. This program still contains a lot of challenging problems for analysts. As the first step, we prove that if a sequence of solutions (u(1k), u(2k)) blows up, then one of h (j) e(ujk)/integral(M)h (j) e(ujk)dv(g), j = 1, 2tends to a sum of Dirac measures. This is so-called the phenomena of weakconcentration. Our purposes in this article are (i) to introduce the shadow system due to the bubbling phenomena when one of parameters rho(i) crosses 4 pi and rho(j) is not an element of 4 pi N where 1 <= i not equal j <= 2; (ii) to show how to calculate the topological degree of Toda systems by computing the topological degree of the generalshadow systems; (iii) to calculate the topological degree of the shadow system for one point blow up. We believe that the degree counting formula for the shadow system would be useful in other problems.