Existence Results of Singular Toda Systems with Sign-Changing Weight Functions

Abstract

We consider the existence problem of the following Singular Toda system on a compact Riemann surface $(\Sigma ,g)$ without boundary \begin{equation*} \begin{cases} -Δ_gu_1=2\overlineρ_1\Big({\frac{h_1e^{u_1}}{\int_Σh_1e^{u_1}dV_g}}-1\Big)-ρ_2\Big({\frac{h_2e^{u_2}}{\int_Σh_2e^{u_2}dV_g}}-1\Big)-4πα_1(δ_0-1), -Δ_gu_2=2ρ_2\big({\frac{h_2e^{u_2}}{\int_Σh_2e^{u_2}dV_g}}-1\big)-\overlineρ_1\big({\frac{h_1e^{u_1}}{\int_Σh_1e^{u_1}dV_g}}-1\big)-4πα_2(δ_0-1), \end{cases} \end{equation*} where $h_1,h_2$ are sign-changing smooth functions, ${\overline{\rho }}_1:=4\pi (1+{\overline{\alpha }}_1),0<{\rho }_2<4\pi (1+{\overline{\alpha }}_2),{\overline{\alpha }}_i=\min \{0,{\alpha }_i\},{\alpha }_i>-1,i=1,2$. By relying on the proof framework established in \cite{DJLW}, the Pohozaev identity and the classical blow-up analysis, we prove the existence theorem under some appropriate condition. Our results generalize Jost-Wang's results \cite{JLW} from regular Toda system with positive functions to the singular Toda system involving sign-changing weight functions.

22 pages