Existence results for Toda systems with sign-changing prescribed functions: Part I
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912147803471872 |
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| author | Sun, LinLin Zhu, Xiaobao |
| author_facet | Sun, LinLin Zhu, Xiaobao |
| contents | Let $(M,g)$ be a compact Riemann surface with area $1$, we shall study the Toda system $$
\begin{cases}
-Δu_1 = 2ρ_1(h_1e^{u_1}-1) - ρ_2(h_2e^{u_2}-1),\\
-Δu_2 = 2ρ_2(h_2e^{u_2}-1) - ρ_1(h_1e^{u_1}-1),
\end{cases} $$ on $(M,g)$ with $ρ_1=4π$, $ρ_2\in(0,4π)$, $h_1$ and $h_2$ are two smooth functions on $M$. In Jost-Lin-Wang's celebrated article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), they obtained a sufficient condition for the existence of this Toda system when $h_1$ and $h_2$ are both positive. In this paper, we shall improve this result to the case $h_1$ and $h_2$ can change signs. We shall pursue a variational method and use the standard blowup analysis. Among other things, the main contribution in our proof is to show the blowup can only happen at one point where $h_1$ is positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence results for Toda systems with sign-changing prescribed functions: Part I Sun, LinLin Zhu, Xiaobao Analysis of PDEs Differential Geometry Let $(M,g)$ be a compact Riemann surface with area $1$, we shall study the Toda system $$ \begin{cases} -Δu_1 = 2ρ_1(h_1e^{u_1}-1) - ρ_2(h_2e^{u_2}-1),\\ -Δu_2 = 2ρ_2(h_2e^{u_2}-1) - ρ_1(h_1e^{u_1}-1), \end{cases} $$ on $(M,g)$ with $ρ_1=4π$, $ρ_2\in(0,4π)$, $h_1$ and $h_2$ are two smooth functions on $M$. In Jost-Lin-Wang's celebrated article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), they obtained a sufficient condition for the existence of this Toda system when $h_1$ and $h_2$ are both positive. In this paper, we shall improve this result to the case $h_1$ and $h_2$ can change signs. We shall pursue a variational method and use the standard blowup analysis. Among other things, the main contribution in our proof is to show the blowup can only happen at one point where $h_1$ is positive. |
| title | Existence results for Toda systems with sign-changing prescribed functions: Part I |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2412.05578 |