<p>We study positivity properties of the quasilinear elliptic equation <Equation ID="Equ7"> <EquationSource Format="TEX">$\begin{aligned} -\textrm{div}\mathcal {A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty )\qquad \text{ in } \Omega , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mtext>div</mtext> <mi mathvariant="script">A</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mi>V</mi> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="2em" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where the function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {A}(x,\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is induced by a family of norms on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) parameterized by points in the domain <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega \subseteq \mathbb {R}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, and <i>V</i> belongs to a certain local Morrey space. We first establish some two-sided estimates for the Bregman distances of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|\xi |^{p}_{s,a}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">|</mo> <mi>ξ</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>a</mi> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1<s<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>s</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>), where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$a=(a_{1},a_{2},\ldots ,a_{n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$a_{1},a_{2},\ldots ,a_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> are certain functions with positive local lower and upper bounds in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. These estimates lead to a Maz’ya-type characterization for Hardy-weights of the corresponding functionals. Then we prove three types of sufficient conditions for the attainment of the Hardy constant in a certain space <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\widetilde{W}^{1,p}_{0}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mover accent="true"> <mi>W</mi> <mo stretchy="false">~</mo> </mover> <mn>0</mn> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

期刊论文
出版模式：

开放获取
发表日期：

2026年7月6日

查看全文

Yongjun Hou

摘要

We study positivity properties of the quasilinear elliptic equation $\begin{aligned} -\textrm{div}\mathcal {A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty )\qquad \text{ in } \Omega , \end{aligned}$ $\begin{matrix} - div A (x, \nabla u) + {V | u |}^{p - 2} u = 0 (1 < p < \infty) in Ω, \end{matrix}$ where the function $\mathcal {A}(x,\xi )$ $A (x, ξ)$ is induced by a family of norms on $\mathbb {R}^{n}$ $R^{n}$ ( $n\ge 2$ $n \geq 2$ ) parameterized by points in the domain $\Omega \subseteq \mathbb {R}^{n}$ $Ω \subseteq R^{n}$ , and V belongs to a certain local Morrey space. We first establish some two-sided estimates for the Bregman distances of $|\xi |^{p}_{s,a}$ ${| ξ |}_{s, a}^{p}$ ( $1<s<\infty $ $1 < s < \infty$ ), where $a=(a_{1},a_{2},\ldots ,a_{n})$ $a = (a_{1}, a_{2}, \dots, a_{n})$ and $a_{1},a_{2},\ldots ,a_{n}$ $a_{1}, a_{2}, \dots, a_{n}$ are certain functions with positive local lower and upper bounds in $\Omega $ $Ω$ . 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 $\widetilde{W}^{1,p}_{0}(\Omega )$ ${\tilde{W}}_{0}^{1, p} (Ω)$ .