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<p>In this paper, we investigate some quantitative properties for positive smooth solutions of the following semilinear elliptic equation involving the weighted Laplacian <Equation ID="Equ45"> <EquationSource Format="TEX">\(\begin{aligned} \Delta _{\phi } u+f(u)=0 \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> </msub> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>on a complete weighted manifold with the Dirichlet boundary condition. We obtain a Yau-type gradient estimate for positive smooth solutions of the equation when the <i>m</i>-Bakry–Émery Ricci tensor and weighted mean curvature are bounded from below. Based on this estimate, we can derive, among other things, Harnack inequalities, Liouville-type and non-existence results. Some applications of these results for specific partial differential equations that arise in geometry and physics are presented and discussed.</p>

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Some quantitative properties of positive smooth solutions to semilinear elliptic equations on weighted manifolds with compact boundary and applications

  • 期刊论文

  • 发表日期：

    2026年5月7日

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  • Ha Tuan Dung

摘要

In this paper, we investigate some quantitative properties for positive smooth solutions of the following semilinear elliptic equation involving the weighted Laplacian \(\begin{aligned} \Delta _{\phi } u+f(u)=0 \end{aligned}\) Δ ϕ u + f ( u ) = 0 on a complete weighted manifold with the Dirichlet boundary condition. We obtain a Yau-type gradient estimate for positive smooth solutions of the equation when the m-Bakry–Émery Ricci tensor and weighted mean curvature are bounded from below. Based on this estimate, we can derive, among other things, Harnack inequalities, Liouville-type and non-existence results. Some applications of these results for specific partial differential equations that arise in geometry and physics are presented and discussed.