发表机构
Faculty of Economic Mathematics, University of Economics and Law; Vietnam National University（经济数学学院，经济与法律大学; 越南国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在无有限能量假设下完整分类了临界各向异性 p-Laplace 方程的非负局部弱解，证明所有非零解均为各向异性 Sobolev 气泡，并建立了 Schoen 型 Harnack 不等式。
AI 中文摘要
设 $N\ge 2$ 且 $1<p<N$。令 $H$ 为 $\mathbb{R}^N$ 上可能不可逆的 Minkowski 范数，满足 $H\in C^2(\mathbb{R}^N\setminus\{0\})$ 且 $D^2(H^2/2)$ 在远离原点处一致正定。我们在没有任何有限能量、可积性或渐近假设的条件下，分类了方程 \\[ -\operatorname{div}\big(H(\nabla u)^{p-1}\nabla H(\nabla u)\big)=u^{p^*-1},\qquad p^*=\frac{Np}{N-p} \\] 的所有非负局部弱解 $u\in W^{1,p}_{\rm loc}(\mathbb{R}^N)$。每个非零解都是各向异性 Sobolev 气泡。这解决了 Antonini--Ciraolo--Gatti 问题 A 的各向异性类比（其欧几里得情形由 Zhang 解决），覆盖了完整范围 $1<p<N$，并去除了 Chen--Wu--Yan--Yang 的非能量各向异性分类中对 $p$ 的限制。我们还证明了局部解的尺度不变 Schoen 型 Harnack 不等式。证明遵循 Zhang 处理欧几里得 $p$-Laplace 算子的策略，将有界爆破轮廓的刚性与首次接触论证以及 Ciraolo 和 Gatti 的积分自举相结合。各向异性分析需要两个额外要素。首先，我们在非临界集上以分布形式建立了共轭应力张量的 Bochner 恒等式，仅使用 $H$ 的 $C^2$ 正则性。其次，我们证明了 Finsler $p$-调和函数在孤立奇点处的两项展开，包括 Pohozaev 不变量所需的梯度控制。对于不可逆各向异性，论证使用反射范数表述。直接为相关 Leray--Lions 算子证明的 Morrey 型估计取代了欧几里得论证中使用的位势理论输入。
英文摘要
Let $N\ge 2$ and $1<p<N$. Let $H$ be a possibly non-reversible Minkowski norm on $\mathbb{R}^N$, with $H\in C^2(\mathbb{R}^N\setminus\{0\})$ and $D^2(H^2/2)$ uniformly positive definite away from the origin. We classify all nonnegative local weak solutions $u\in W^{1,p}_{\rm loc}(\mathbb{R}^N)$ of \[ -\operatorname{div}\big(H(\nabla u)^{p-1}\nabla H(\nabla u)\big)=u^{p^*-1},\qquad p^*=\frac{Np}{N-p}, \] without any finite-energy, integrability, or asymptotic assumption. Every nonzero solution is an anisotropic Sobolev bubble. This settles, for the full range $1<p<N$, the anisotropic analogue of Problem A of Antonini--Ciraolo--Gatti, whose Euclidean case was settled by Zhang, and removes the restriction on $p$ in the non-energy anisotropic classification of Chen--Wu--Yan--Yang. We also prove a scale-invariant Schoen-type Harnack inequality for local solutions. The proof follows the strategy of Zhang for the Euclidean $p$-Laplacian, combining rigidity of bounded blow-up profiles and a first-contact argument with the integral bootstrap of Ciraolo and Gatti. The anisotropic analysis requires two additional ingredients. First, we establish a Bochner identity for the conjugated stress tensor in distributional form on the noncritical set, using only $C^2$ regularity of $H$. Second, we prove a two-term expansion at isolated singularities of Finsler $p$-harmonic functions, including the gradient control needed by the Pohozaev invariant. For non-reversible anisotropies the argument is formulated with the reflected norm. A Morrey-type estimate proved directly for the relevant Leray--Lions operator replaces the potential-theoretic input used in the Euclidean argument.
Comments24 pages
