Degree counting for Toda system with simple singularity: One point blow up

Abstract

In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when rho(1) is an element of(0, 4 pi) boolean OR(4 pi, 8 pi) and rho(2) is not an element of 4 pi N. The key step is to derive the degree formula of the shadow system, which arises from the bubbling solutions as rho(1) tends to 4 pi. In order to compute the topological degree of the shadow system, we need to find some suitable deformation. During this deformation, we shall deal with new difficulty arising from the phenomenon: blow up does not necessarily imply concentration of mass. This phenomenon occurs due to the collapsing of singularities. This is a continuation of the previous work [25].