Existence results for Toda systems with sign-changing prescribed functions: Part II
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2024
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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 investigate the Toda system \begin{align} \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} \end{align} on $(M, g)$ where $ρ_1, ρ_2 \in (0,4π]$, and $h_1$ and $h_2$ are two smooth functions on $M$.When some $ρ_i$ equals $4π$, the Toda system becomes critical with respect to the Moser-Trudinger inequality for it, making the existence problem significantly more challenging. In their seminal article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), Jost, Lin, and Wang established sufficient conditions for the existence of solutions the Toda system when $ρ_1=4π$, $ρ_2 \in (0,4π)$ or $ρ_1=ρ_2=4π$, assuming that $h_1$ and $h_2$ are both positive. In our previous paper we extended these results to allow $h_1$ and $h_2$ to change signs in the case $ρ_1=4π$, $ρ_2 \in (0,4π)$. In this paper we further extend the study to prove that Jost-Lin-Wang's sufficient conditions remain valid even when $h_1$ and $h_2$ can change signs and $ρ_1=ρ_2=4π$. Our proof relies on an improved version of the Moser-Trudinger inequality for the Toda system, along with edicated analyses similar to Brezis-Merle type and the use of Pohozaev identities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_07537 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence results for Toda systems with sign-changing prescribed functions: Part II Sun, Linlin Zhu, Xiaobao Analysis of PDEs Differential Geometry Let $(M, g)$ be a compact Riemann surface with area $1$. We investigate the Toda system \begin{align} \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} \end{align} on $(M, g)$ where $ρ_1, ρ_2 \in (0,4π]$, and $h_1$ and $h_2$ are two smooth functions on $M$.When some $ρ_i$ equals $4π$, the Toda system becomes critical with respect to the Moser-Trudinger inequality for it, making the existence problem significantly more challenging. In their seminal article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), Jost, Lin, and Wang established sufficient conditions for the existence of solutions the Toda system when $ρ_1=4π$, $ρ_2 \in (0,4π)$ or $ρ_1=ρ_2=4π$, assuming that $h_1$ and $h_2$ are both positive. In our previous paper we extended these results to allow $h_1$ and $h_2$ to change signs in the case $ρ_1=4π$, $ρ_2 \in (0,4π)$. In this paper we further extend the study to prove that Jost-Lin-Wang's sufficient conditions remain valid even when $h_1$ and $h_2$ can change signs and $ρ_1=ρ_2=4π$. Our proof relies on an improved version of the Moser-Trudinger inequality for the Toda system, along with edicated analyses similar to Brezis-Merle type and the use of Pohozaev identities. |
| title | Existence results for Toda systems with sign-changing prescribed functions: Part II |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2412.07537 |