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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.08914 |
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Table of Contents:
- We consider the existence problem of the following Singular Toda system on a compact Riemann surface $(Σ, g)$ 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, $\overlineρ_1:=4π(1+\overlineα_1),\,0<ρ_2<4π(1+\overlineα_2),\,\overlineα_i=\min\{0,α_i\},\,α_i>-1,\,i=1,2$. 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.