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

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 (u1k,u2k) blows up, then one of hjeujk∫Mhjeujkdvg, j=1,2 tends to a sum of Dirac measures. This is so-called the phenomena of weak concentration. Our purposes in this article are (i) to introduce the shadow system due to the bubbling phenomena when one of parameters ρi crosses 4π and ρj∉4πN where 1≤i≠j≤2; (ii) to show how to calculate the topological degree of Toda systems by computing the topological degree of the general shadow 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.