In this paper, we first study carefully the positive solutions to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Delta u+\lambda _{1}u\ln u +\lambda _{2}u^{\alpha +1}=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mi>u</mi> <mo>ln</mo> <mi>u</mi> <mo>+</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <msup> <mi>u</mi> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> defined on a complete non-compact Riemannian manifold (M, g) with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Ric(g)\geqslant -Kg$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>i</mi> <mi>c</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>⩾</mo> <mo>-</mo> <mi>K</mi> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation>, which can be regarded as Lichnerowicz-type equations, and according to the different parameter values in the equation, seven cases are discussed to obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on (M, g). For the case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$0<\alpha <\frac{2}{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, this improves considerably the previous related results. Moreover, we also obtain the Liouville-type result for these equations when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Ric(g)\geqslant 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>i</mi> <mi>c</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>⩾</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and establish the Harnack inequality as consequences.

Gradient Estimates for Lichnerowicz-Type Equations

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发表日期：

2025年2月20日

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Xingan Bian,
Pingliang Huang

摘要

In this paper, we first study carefully the positive solutions to $\Delta u+\lambda _{1}u\ln u +\lambda _{2}u^{\alpha +1}=0$ $Δ u + λ_{1} u ln u + λ_{2} u^{α + 1} = 0$ defined on a complete non-compact Riemannian manifold (M, g) with $Ric(g)\geqslant -Kg$ $R i c (g) ⩾ - K g$ , which can be regarded as Lichnerowicz-type equations, and according to the different parameter values in the equation, seven cases are discussed to obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on (M, g). For the case $0<\alpha <\frac{2}{n}$ $0 < α < \frac{2}{n}$ , this improves considerably the previous related results. Moreover, we also obtain the Liouville-type result for these equations when $Ric(g)\geqslant 0$ $R i c (g) ⩾ 0$ and establish the Harnack inequality as consequences.