arXiv subjects

Yongjun Hou

Publications and source records attributed to Yongjun Hou.

4 recordsLinked to original sources

arXiv2026

Fuchsian-type singularity for the Finsler $p$-Laplacian with potential

Let $\Omega\subseteq\mathbb{R}^{n}$ ($2\leq n\in\mathbb{N}$) be a domain and let $\zeta\in\{0,\infty\}$ be an isolated point of the boundary of $\Omega$ in the one-point compactification of $\mathbb{R}^{n}$ with the ideal point $\infty$. Under some further conditions, we study Fuchsian-type singularity at $\zeta$ for the Finsler $p$-Laplace equation with a potential $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{in } \Omega,$$ where $\mathcal{A}(x,\xi)\triangleq\nabla_{\xi}(H(x,\xi)^{p}/p)$ for almost all $x\in\Omega$ and all $\xi\in\mathbb{R}^{n}$, $H$ is a family of norms on $\mathbb{R}^{n}$ ($n\geq 2$) parameterized by points $x\in\Omega$, and $V$ belongs to a local Morrey space. In particular, we investigate asymptotic behaviors of positive solutions of the equation near $\zeta$ and asymptotic behaviors of their quotients.

math.AP↗

arXiv2025

Optimal Hardy-weights for the Finsler $p$-Dirichlet integral with a potential

Fix an integer $n\geq 2$, an exponent $1<p<\infty$, and a domain $\Omega\subseteq\mathbb{R}^{n}$. Let $\Omega^{*}\triangleq\Omega\setminus\{\hat{x}\}$ where $\hat{x}\in\Omega$. Under some further conditions, we construct optimal Hardy-weights for the Finsler $p$-Dirichlet integral $$Q_{0}[\phi;\Omega^{*}]\triangleq\int_{\Omega^{*}}H(x,\nabla \phi)^{p}\,\mathrm{d}x\quad \mbox{on}\quad C^{\infty}_{c}(\Omega^{*}),$$ and the Finsler $p$-Dirichlet integral with a potential $$Q_{V}[\phi;\Omega]\triangleq\int_{\Omega}\left(H(x,\nabla \phi)^{p}+ V|\phi|^{p}\right)\,\mathrm{d}x\quad \mbox{on}\quad C^{\infty}_{c}(\Omega),$$where $H(x,\cdot)$ is a family of norms on $\mathbb{R}^{n}$ parameterized by $x\in\Omega^{*}$ or $x\in\Omega$, respectively, and the potential $V$ lies in a subspace $\widehat{M}^{q}_{{\rm loc}}(p;\Omega)$ of a local Morrey space $M^{q}_{{\rm loc}}(p;\Omega)$.

math.AP↗

arXiv2024

Finsler $p$-Laplace equation with a potential: Maz'ya-type characterization and attainments of the Hardy constant

We study positive properties of the quasilinear elliptic equation $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{ in } \Omega,$$ where the function $\mathcal{A}(x,\xi)$ is induced by a family of norms on $\mathbb{R}^{n}$ ($n\geq 2$) parameterized by points in the domain $\Omega\subseteq\mathbb{R}^{n}$, and $V$ belongs to a certain local Morrey space. We first establish two-sided estimates for Bregman distances of $|\xi|^{p}_{s,a}$ ($1<s<\infty$), where $a=(a_{1},a_{2},\ldots,a_{n})$ and $a_{1},a_{2},\ldots,a_{n}$ are certain functions with positive local lower and upper bounds in $\Omega$. These estimates lead to a Maz'ya-type characterization for Hardy-weights of the corresponding functionals. Then we prove three types of sufficient conditions for the attainment of the Hardy constant in a certain space $\widetilde{W}^{1,p}_{0}(\Omega)$.

math.AP↗

arXiv2021

Positive solutions of the $\mathcal{A}$-Laplace equation with a potential

In this paper, we study positive solutions of the quasilinear elliptic equation $$Q'_{p,\mathcal{A},V}[u]\triangleq-\mathrm{div}{\mathcal{A}(x,\nabla u)}+V(x)|u|^{p-2}u=0,$$ in a domain $\Omega\subseteq \mathbb{R}^n$, where $n\geq 2$, $1<p<\infty$, the divergence of $\mathcal{A}$ is the well known $\mathcal{A}$-Laplace operator considered in the influential book of Heinonen, Kilpel\"{a}inen, and Martio, and the potential $V$ belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator $Q'_{p,\mathcal{A},V}$. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of $Q'_{p,\mathcal{A},V}$ in a domain $\omega\Subset\Omega$, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation $Q'_{p,\mathcal{A},V}[u]=0$ of minimal growth at infinity in $\Omega$, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights.

math.AP↗