Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones
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2024
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| author | Dai, Wei Gui, Changfeng Luo, YunPeng |
| author_facet | Dai, Wei Gui, Changfeng Luo, YunPeng |
| contents | We consider the anisotropic Finsler $N$-Laplacian Liouville equation \[-Δ^{H}_{N}u=e^u \qquad {\rm{in}}\,\, \mathcal{C},\] where $N\geq2$, $\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone including $\mathbb{R}^{N}$, the half space $\mathbb{R}^{N}_{+}$ and $\frac{1}{2^{m}}$-space $\mathbb{R}^{N}_{2^{-m}}:=\{x\in\mathbb{R}^{N}\mid x_{1},\cdots,x_{m}>0\}$ ($m=1,\cdots,N$), and the anisotropic Finsler $N$-Laplacian $Δ^{H}_{N}$ is induced by a positively homogeneous function $H(x)$ of degree $1$. All solutions to the Finsler $N$-Laplacian Liouville equation with finite mass are completely classified. In particular, if $H(ξ)=|ξ|$, then the Finsler $N$-Laplacian $Δ^{H}_{N}$ reduces to the regular $N$-Laplacian $Δ_N$. Our result is a counterpart in the limiting case $p=N$ of the classification results in \cite{CFR} for the critical anisotropic $p$-Laplacian equations with $1<p<N$ in convex cones, and also extends the classification results in \cite{CK,CL,CW,CL2,E} for Liouville equation in the whole space $\mathbb{R}^{N}$ to general convex cones. In our proof, besides exploiting the anisotropic isoperimetric inequality inside convex cones, we have also proved and applied the radial Poincaré type inequality (Lemma \ref{A1}), which are key ingredients in the proof and of their own importance and interests. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones Dai, Wei Gui, Changfeng Luo, YunPeng Analysis of PDEs 35J92, 35B06, 35B40 We consider the anisotropic Finsler $N$-Laplacian Liouville equation \[-Δ^{H}_{N}u=e^u \qquad {\rm{in}}\,\, \mathcal{C},\] where $N\geq2$, $\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone including $\mathbb{R}^{N}$, the half space $\mathbb{R}^{N}_{+}$ and $\frac{1}{2^{m}}$-space $\mathbb{R}^{N}_{2^{-m}}:=\{x\in\mathbb{R}^{N}\mid x_{1},\cdots,x_{m}>0\}$ ($m=1,\cdots,N$), and the anisotropic Finsler $N$-Laplacian $Δ^{H}_{N}$ is induced by a positively homogeneous function $H(x)$ of degree $1$. All solutions to the Finsler $N$-Laplacian Liouville equation with finite mass are completely classified. In particular, if $H(ξ)=|ξ|$, then the Finsler $N$-Laplacian $Δ^{H}_{N}$ reduces to the regular $N$-Laplacian $Δ_N$. Our result is a counterpart in the limiting case $p=N$ of the classification results in \cite{CFR} for the critical anisotropic $p$-Laplacian equations with $1<p<N$ in convex cones, and also extends the classification results in \cite{CK,CL,CW,CL2,E} for Liouville equation in the whole space $\mathbb{R}^{N}$ to general convex cones. In our proof, besides exploiting the anisotropic isoperimetric inequality inside convex cones, we have also proved and applied the radial Poincaré type inequality (Lemma \ref{A1}), which are key ingredients in the proof and of their own importance and interests. |
| title | Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones |
| topic | Analysis of PDEs 35J92, 35B06, 35B40 |
| url | https://arxiv.org/abs/2407.04987 |