By virtue of the Nash–Moser iteration, we obtain a local gradient estimate of positive weak solutions to the weighted p-Laplacian equation <Equation ID="Equ38"> <EquationSource Format="TEX">$\begin{aligned} \Delta _{p,f}u+a(x)u\ln u=0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mi>u</mi> <mo>+</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>ln</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>defined on a complete smooth metric measure space under the condition that the m-Bakry–Émery Ricci curvature has a lower bound, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a(x)\le 0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> As applications, Liouville-type theorems for positive solutions to the above equation are achieved.

Gradient estimates for a class of p-Laplacian equations

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发表日期：

2025年11月14日

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Guangyue Huang,
Jingxu Liu,
Zhen Wang

摘要

By virtue of the Nash–Moser iteration, we obtain a local gradient estimate of positive weak solutions to the weighted p-Laplacian equation $\begin{aligned} \Delta _{p,f}u+a(x)u\ln u=0 \end{aligned}$ $\begin{matrix} Δ_{p, f} u + a (x) u ln u = 0 \end{matrix}$ defined on a complete smooth metric measure space under the condition that the m-Bakry–Émery Ricci curvature has a lower bound, where $p>2$ $p > 2$ and the function $a(x)\le 0.$ $a (x) \leq 0 .$ As applications, Liouville-type theorems for positive solutions to the above equation are achieved.