In this paper, we combine Bochner formula, Saloff-Coste’s Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Delta _pu+\Delta _qu+h(u,|\nabla u|^2)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>q</mi> </msub> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation> defined on a complete Riemannian manifold <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\left( M,g\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>M</mi> <mo>,</mo> <mi>g</mi> </mfenced> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$q\ge p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$h\in C^1(\mathbb {R}\times \mathbb {R}^{+})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>∈</mo> <msup> <mi>C</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Delta _z u=\textrm{div}\left( \left| \nabla u\right| ^{z-2}\nabla u\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>z</mi> </msub> <mi>u</mi> <mo>=</mo> <mtext>div</mtext> <mfenced close=")" open="("> <msup> <mfenced close="|" open="|"> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mfenced> <mrow> <mi>z</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$z\in \{ p,~q\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mi>p</mi> <mo>,</mo> <mspace width="3.33333pt" /> <mi>q</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, is the usual z-Laplace operator. Under some assumptions on p, q and h(x, y), we derive concise gradient estimates for solutions to the above equation and then obtain some Liouville type theorems. In particular, we use integral estimate method to show that, if u is a non-negative entire solution to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Delta _p u +\Delta _q u=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>q</mi> </msub> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\le p\le q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>) on a complete non-compact Riemannian manifold M with non-negative Ricci curvature and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\dim M = n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>dim</mo> <mi>M</mi> <mo>=</mo> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, then u is a trivial constant solution.

Liouville Type Theorems for Some (p, q)-Laplace Equations with Gradient Dependent Reaction on Riemannian Manifolds

期刊论文
发表日期：

2026年5月5日

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Youde Wang,
Liqin Zhang

摘要

In this paper, we combine Bochner formula, Saloff-Coste’s Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation $\Delta _pu+\Delta _qu+h(u,|\nabla u|^2)=0$ $Δ_{p} u + Δ_{q} u + {h (u, | \nabla u |}^{2}) = 0$ defined on a complete Riemannian manifold $\left( M,g\right) $ $(M, g)$ , where $q\ge p>1$ $q \geq p > 1$ , $h\in C^1(\mathbb {R}\times \mathbb {R}^{+})$ $h \in C^{1} (R \times R^{+})$ and $\Delta _z u=\textrm{div}\left( \left| \nabla u\right| ^{z-2}\nabla u\right) $ $Δ_{z} u = div ({|\nabla, u|}^{z - 2}, \nabla, u)$ , with $z\in \{ p,~q\}$ $z \in {p, q}$ , is the usual z-Laplace operator. Under some assumptions on p, q and h(x, y), we derive concise gradient estimates for solutions to the above equation and then obtain some Liouville type theorems. In particular, we use integral estimate method to show that, if u is a non-negative entire solution to $\Delta _p u +\Delta _q u=0$ $Δ_{p} u + Δ_{q} u = 0$ ( $n\le p\le q$ $n \leq p \leq q$ ) on a complete non-compact Riemannian manifold M with non-negative Ricci curvature and $\dim M = n\ge 2$ $dim M = n \geq 2$ , then u is a trivial constant solution.