Degree Counting Theorems for 2x2 non-symmetric singular Liouville Systems

Abstract

Let $(M,g)$ be a compact Riemann surface with no boundary and $u=(u_1,u_2)$ be a solution of the following singular Liouville system: $${\Delta }_g{u}_i+\sum _{j=1}^2{a}_{ij}{\rho }_j(\frac{h_j{e}^{u_j}}{{\int }_M{h}_j{e}^{u_j}d{V}_g}-1)=\sum _{l=1}^{N}4\pi {\gamma }_l({\delta }_{p_l}-1),$$ where $h_1,h_2$ are positive smooth functions, $p_1,\cdots ,p_N$ are distinct points on $M$, ${\delta }_{p_l}$ are Dirac masses, $\rho =({\rho }_1,{\rho }_2)({\rho }_i\ge 0)$ and $({\gamma }_1,\cdots ,{\gamma }_N)({\gamma }_l>-1)$ are constant vectors. In the previous work, we derive a degree counting formula for the singular Liouville system when $A$ satisfies standard assumptions. In this article, we establish a more general degree counting formula for 2$\times$2 singular Liouville system when the coefficient matrix $A$ is non-symmetric and non-invertible. Finally, the existence of solution can be proved by the degree counting formula which depends only on the topology of the domain and the location of $\rho$.

13 pages