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<p>In this paper, we use the Nash-Moser iteration method to prove the local and global gradient estimates of positive solutions to the weighted <i>p</i>-Laplacian equation <Equation ID="Equ76"> <EquationSource Format="TEX">\(\begin{aligned} \Delta _{p,f} u+au^\sigma =0 \end{aligned}\)</EquationSource> </Equation>defined on a complete smooth metric measure space under the condition that the <i>m</i>-Bakry-Émery Ricci curvature has a lower bound, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p&gt;1\)</EquationSource> </InlineEquation>, <i>a</i> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sigma \in \mathbb {R}\)</EquationSource> </InlineEquation> are constants and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Delta _{p,f} u=e^{f}\textrm{div}(e^{-f}|\nabla u|^{p-2}\nabla u)\)</EquationSource> </InlineEquation> is the weighted <i>p</i>-Laplacian operator. As applications, we derive Liouville type theorems and Harnack inequalities for positive solutions to the above equation.</p>

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Local and Global Gradient Estimates for p-Laplace Equations on Complete Smooth Metric Measure Spaces and Liouville Type Theorems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月29日

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  • Guangyue Huang,
  • Jingxu Liu

摘要

In this paper, we use the Nash-Moser iteration method to prove the local and global gradient estimates of positive solutions to the weighted p-Laplacian equation \(\begin{aligned} \Delta _{p,f} u+au^\sigma =0 \end{aligned}\) defined on a complete smooth metric measure space under the condition that the m-Bakry-Émery Ricci curvature has a lower bound, where \(p>1\) , a and \(\sigma \in \mathbb {R}\) are constants and \(\Delta _{p,f} u=e^{f}\textrm{div}(e^{-f}|\nabla u|^{p-2}\nabla u)\) is the weighted p-Laplacian operator. As applications, we derive Liouville type theorems and Harnack inequalities for positive solutions to the above equation.