On the classification of solutions to a class of $N$-Liouville equations in $\mathbb{R}^N$
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arXiv
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| Format: | Preprint |
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2026
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| author | Ciraolo, Giulio Esposito, Pierpaolo Li, Xiaoliang |
| author_facet | Ciraolo, Giulio Esposito, Pierpaolo Li, Xiaoliang |
| contents | Given $N\geq 2$ and $α>-1$, we consider the following weighted Liouville-type equation involving the $N$-Laplacian: \begin{equation*} \left\{ \begin{aligned} -& Δ_N u = |x|^{Nα} e^u \quad \text{ in } \mathbb{R}^N && , \\ & \int_{\mathbb{R}^N} |x|^{Nα} e^u \, dx < + \infty\,. &&\end{aligned} \right. \end{equation*}
Solutions have been completely classified when $N=2$ via complex analysis, and when $α=0$ using Pohozaev identities and an isoperimetric argument.
In this paper, we first devise a $P$-function approach to the classification result for all $α>-1$ when $N=2$. Since it is not based on complex analysis, this alternative and more PDE-oriented approach naturally extends to $N\geq 3$ by providing the classification for any $-1<α\leq 0$. In particular, the explicit radial solutions are the unique ones for $-1<α\leq0$ but become degenerate for special values $α_k>0$, a hint that non-radial solutions might arise for $α>0$ as it happens when $N=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10050 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the classification of solutions to a class of $N$-Liouville equations in $\mathbb{R}^N$ Ciraolo, Giulio Esposito, Pierpaolo Li, Xiaoliang Analysis of PDEs Given $N\geq 2$ and $α>-1$, we consider the following weighted Liouville-type equation involving the $N$-Laplacian: \begin{equation*} \left\{ \begin{aligned} -& Δ_N u = |x|^{Nα} e^u \quad \text{ in } \mathbb{R}^N && , \\ & \int_{\mathbb{R}^N} |x|^{Nα} e^u \, dx < + \infty\,. &&\end{aligned} \right. \end{equation*} Solutions have been completely classified when $N=2$ via complex analysis, and when $α=0$ using Pohozaev identities and an isoperimetric argument. In this paper, we first devise a $P$-function approach to the classification result for all $α>-1$ when $N=2$. Since it is not based on complex analysis, this alternative and more PDE-oriented approach naturally extends to $N\geq 3$ by providing the classification for any $-1<α\leq 0$. In particular, the explicit radial solutions are the unique ones for $-1<α\leq0$ but become degenerate for special values $α_k>0$, a hint that non-radial solutions might arise for $α>0$ as it happens when $N=2$. |
| title | On the classification of solutions to a class of $N$-Liouville equations in $\mathbb{R}^N$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.10050 |