In this paper, we provide a new approach to employ the Nash–Moser iteration technique to analyze the local and global properties of positive solutions to the equation <Equation ID="Equ102"> <EquationSource Format="TEX">$\begin{aligned} \Delta _pv + a|\nabla v|^qv^r =0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>v</mi> <mo>+</mo> <mi>a</mi> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> </msup> <msup> <mi>v</mi> <mi>r</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>on a complete Riemannian manifold with Ricci curvature bounded from below, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and q, r, and a are real constants. Assuming certain conditions on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a,\, p,\, q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>p</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> and r, we can derive universal and succinct Cheng–Yau type logarithmic gradient estimates for such solutions. In particular, we give the explicit expressions of constants in the logarithmic gradient estimate for entire solutions to the above equation (see Theorem <InternalRef RefID="FPar18">1.11</InternalRef>). The gradient estimates enable us to obtain some Liouville-type theorems, Harnack inequalities, and local estimates near singularities for positive solutions. Some of our results are new even in the case where the domain is a Euclidean space and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Nash–Moser iteration approach to the logarithmic gradient estimates and Liouville properties of quasilinear elliptic equations on manifolds

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发表日期：

2026年4月22日

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Jie He,
Jingchen Hu,
Youde Wang

摘要

In this paper, we provide a new approach to employ the Nash–Moser iteration technique to analyze the local and global properties of positive solutions to the equation $\begin{aligned} \Delta _pv + a|\nabla v|^qv^r =0 \end{aligned}$ $\begin{matrix} Δ_{p} v + a {| \nabla v |}^{q} v^{r} = 0 \end{matrix}$ on a complete Riemannian manifold with Ricci curvature bounded from below, where $p>1$ $p > 1$ , and q, r, and a are real constants. Assuming certain conditions on $a,\, p,\, q$ $a, p, q$ and r, we can derive universal and succinct Cheng–Yau type logarithmic gradient estimates for such solutions. In particular, we give the explicit expressions of constants in the logarithmic gradient estimate for entire solutions to the above equation (see Theorem 1.11). The gradient estimates enable us to obtain some Liouville-type theorems, Harnack inequalities, and local estimates near singularities for positive solutions. Some of our results are new even in the case where the domain is a Euclidean space and $p=2$ $p = 2$ .