Classification of solutions to the anisotropic $N$-Liouville equation in $\mathbb{R}^N$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916260228366336 |
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| author | Ciraolo, Giulio Li, Xiaoliang |
| author_facet | Ciraolo, Giulio Li, Xiaoliang |
| contents | Given $N\geq 2$, we completely classify the solutions of the anisotropic $N$-Liouville equation $$-Δ_N^H\,u=e^u \quad\text{in }\mathbb{R}^N,$$ under the finite mass condition $\int_{\mathbb{R}^N} e^u\,dx<+\infty$. Here $Δ_N^H$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence for $N=2$, we give an affirmative answer to a conjecture made in [G. Wang and C. Xia, J. Differential Equations 252 (2012) 1668--1700]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12039 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classification of solutions to the anisotropic $N$-Liouville equation in $\mathbb{R}^N$ Ciraolo, Giulio Li, Xiaoliang Analysis of PDEs Given $N\geq 2$, we completely classify the solutions of the anisotropic $N$-Liouville equation $$-Δ_N^H\,u=e^u \quad\text{in }\mathbb{R}^N,$$ under the finite mass condition $\int_{\mathbb{R}^N} e^u\,dx<+\infty$. Here $Δ_N^H$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence for $N=2$, we give an affirmative answer to a conjecture made in [G. Wang and C. Xia, J. Differential Equations 252 (2012) 1668--1700]. |
| title | Classification of solutions to the anisotropic $N$-Liouville equation in $\mathbb{R}^N$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2306.12039 |