<p>In this paper we derive local gradient estimate for positive solutions of the weighted nonlinear elliptic equation <Equation ID="Equ40"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40863_2025_495_Article_Equ40.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="212" /> </MediaObject> <EquationSource Format="TEX">$\Delta _{f,p} u+au^\alpha (\ln u)^\beta +bu^\gamma =0,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mi>u</mi> <mo>+</mo> <mi>a</mi> <msup> <mi>u</mi> <mi>α</mi> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mo>ln</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mi>β</mi> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>u</mi> <mi>γ</mi> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40863_2025_495_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="142" /> </InlineMediaObject> <EquationSource Format="TEX">$a\ge 0,b\ge 0,\alpha ,\beta ,\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> </mrow> </math></EquationSource> </InlineEquation> are real constants. We have taken the lower bound assumption on Bakry-Émery Ricci tensor on a complete weighted Riemannian manifold with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40863_2025_495_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$|\nabla f|\le C_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. We have applied our results to provide global gradient estimate for the same equation, derived a Harnack type inequality and established a Liouville type theorem. </p>

Liouville type theorem for weighted p-Laplacian and elliptic gradient estimate

期刊论文
发表日期：

2025年4月4日

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Shyamal Kumar Hui,
Abimbola Abolarinwa,
Sujit Bhattacharyya

摘要

In this paper we derive local gradient estimate for positive solutions of the weighted nonlinear elliptic equation $\Delta _{f,p} u+au^\alpha (\ln u)^\beta +bu^\gamma =0,$ $Δ_{f, p} u + a u^{α} {(ln u)}^{β} + b u^{γ} = 0,$ where $a\ge 0,b\ge 0,\alpha ,\beta ,\gamma $ $a \geq 0, b \geq 0, α, β, γ$ are real constants. We have taken the lower bound assumption on Bakry-Émery Ricci tensor on a complete weighted Riemannian manifold with $|\nabla f|\le C_1$ $| \nabla f | \leq C_{1}$ . We have applied our results to provide global gradient estimate for the same equation, derived a Harnack type inequality and established a Liouville type theorem.