Let M be a complete non-compact Riemannian manifold and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> be a Radon measure on M. We study the existence and non-existence of positive solutions to a nonlocal elliptic inequality <Equation ID="Equ67"> <EquationSource Format="TEX">$\begin{aligned} (-\Delta )^{\alpha } u\ge u^{q}\sigma \quad \text {in}\,\,M, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>α</mi> </msup> <mi>u</mi> <mo>≥</mo> <msup> <mi>u</mi> <mi>q</mi> </msup> <mi>σ</mi> <mspace width="1em" /> <mtext>in</mtext> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi>M</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$q>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. When the Green function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G^{(\alpha )}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> of the fractional Laplacian <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(-\Delta )^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation> exists and satisfies the quasi-metric property, we obtain necessary and sufficient criteria for existence of positive solutions. In particular, explicit conditions in terms of volume growth and the growth of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> are given, when M admits Li-Yau Gaussian type heat kernel estimates.

Sharp criteria for nonlocal elliptic inequalities on manifolds

期刊论文
发表日期：

2026年2月13日

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Qingsong Gu,
Xueping Huang,
Yuhua Sun

摘要

Let M be a complete non-compact Riemannian manifold and $\sigma $ $σ$ be a Radon measure on M. We study the existence and non-existence of positive solutions to a nonlocal elliptic inequality $\begin{aligned} (-\Delta )^{\alpha } u\ge u^{q}\sigma \quad \text {in}\,\,M, \end{aligned}$ $\begin{matrix} {(- Δ)}^{α} u \geq u^{q} σ in M, \end{matrix}$ with $q>1$ $q > 1$ . When the Green function $G^{(\alpha )}$ $G^{(α)}$ of the fractional Laplacian $(-\Delta )^{\alpha }$ ${(- Δ)}^{α}$ exists and satisfies the quasi-metric property, we obtain necessary and sufficient criteria for existence of positive solutions. In particular, explicit conditions in terms of volume growth and the growth of $\sigma $ $σ$ are given, when M admits Li-Yau Gaussian type heat kernel estimates.