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Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones

Wei Dai, Changfeng Gui, YunPeng Luo

TL;DR

The paper classifies all finite-mass solutions to the anisotropic Finsler $N$-Laplacian Liouville equation $- abla^H_Nu=e^u$ in convex cones, establishing a complete Liouville-type profile up to translations and dilations. It introduces and leverages a radial Poincaré type inequality inside cones and an anisotropic isoperimetric framework to obtain sharp asymptotics, mass quantization, and symmetry. Second-order regularity for weak solutions is proved via cone-approximation, enabling a Pohozaev-type analysis that identifies the exact mass and enforces equality in the isoperimetric inequality. The results extend the CFR-type classifications from the subcritical $1<p<N$ setting to the limiting case $p=N$, and generalize Liouville-type classifications from the whole space to general convex cones, with implications for Wulff-type variational problems in anisotropic media.

Abstract

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.

Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones

TL;DR

The paper classifies all finite-mass solutions to the anisotropic Finsler -Laplacian Liouville equation in convex cones, establishing a complete Liouville-type profile up to translations and dilations. It introduces and leverages a radial Poincaré type inequality inside cones and an anisotropic isoperimetric framework to obtain sharp asymptotics, mass quantization, and symmetry. Second-order regularity for weak solutions is proved via cone-approximation, enabling a Pohozaev-type analysis that identifies the exact mass and enforces equality in the isoperimetric inequality. The results extend the CFR-type classifications from the subcritical setting to the limiting case , and generalize Liouville-type classifications from the whole space to general convex cones, with implications for Wulff-type variational problems in anisotropic media.

Abstract

We consider the anisotropic Finsler -Laplacian Liouville equation where , is an open convex cone including , the half space and -space (), and the anisotropic Finsler -Laplacian is induced by a positively homogeneous function of degree . All solutions to the Finsler -Laplacian Liouville equation with finite mass are completely classified. In particular, if , then the Finsler -Laplacian reduces to the regular -Laplacian . Our result is a counterpart in the limiting case of the classification results in \cite{CFR} for the critical anisotropic -Laplacian equations with in convex cones, and also extends the classification results in \cite{CK,CL,CW,CL2,E} for Liouville equation in the whole space 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.
Paper Structure (9 sections, 20 theorems, 251 equations, 1 figure)

This paper contains 9 sections, 20 theorems, 251 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and setting of the problem
  3. Main results
  4. Preliminaries
  5. Sharp asymptotic estimates at infinity on $u$ and $\nabla u$
  6. Second-order regularity for weak solutions
  7. Quantization results by the anisotropic isoperimetric inequality inside convex cones and the Pohozaev type identity
  8. Proof of our main Theorem 1.1
  9. The radial Poincaré type inequality and its applications

Key Result

Theorem 1.1

Let $N\geq2$ and $\mathcal{C}$ be a convex cone in $\mathbb{R}^{N}$ and write $\mathcal{C}=\mathbb{R}^{k}\times\mathcal{\widetilde{C}}$, where $k\in\{0,\cdots ,N\}$ and $\mathcal{\widetilde{C}}\subset\mathbb{R}^{N-k}$ is an open convex cone with vertex at the origin $0_{\mathbb{R}^{N-k}}$ which does for some $\lambda>0$ and $x_0\in\overline{\mathcal{C}}$, where with $c_N:=N\left(\frac{N^2}{N-1}\r

Figures (1)

  • Figure 1: The (front) radial contact set $\Gamma^{-}_{P}$ and (back) radial contact set $\Gamma^{+}_{P}$ of $\Omega$ in X-rays from the radial center $P$

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5: Logarithmic asymptotic behavior in convex cone $\mathcal{C}$ at infinity
  • proof
  • Theorem 2.6
  • ...and 26 more