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<p>We study nonnegative solutions to the following higher-order fractional Yamabe and Hardy-Hénon equations: <Equation ID="Equ1"> <EquationSource Format="TEX">\((-\Delta)^{\sigma}u=|x|^{-\alpha}u^{p}\quad \text{in}\ \mathbb{R}^{n}\backslash \{0\},\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mi>σ</mi> </mrow> </msup> <mi>u</mi> <mo>=</mo> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>−</mo> <mi>α</mi> </mrow> </msup> <msup> <mi>u</mi> <mrow> <mi>p</mi> </mrow> </msup> <mspace width="1em" /> <mtext>in</mtext> <mspace width="thinmathspace" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </math></EquationSource> </Equation> where 0 &lt; <i>σ</i> &lt; <i>n</i>/2, −∞ &lt; <i>α</i> &lt; 2<i>σ, p</i> &gt; 1, and the origin may be a singularity. When the singularity is removable, we establish optimal Liouville-type theorems for all <i>σ</i> ∈ (0, <i>n</i>/2). Moreover, we prove the existence of positive solutions in the supercritical case. For non-removable singularities, we demonstrate the radial symmetry of solutions, which is useful in studying the higher-order fractional singular Yamabe problem.</p>

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Liouville-type theorems and radial symmetry for the Yamabe and Hardy-Hénon equations involving higher-order fractional Laplacians

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  • 发表日期：

    2026年7月22日

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  • Hui Yang

摘要

We study nonnegative solutions to the following higher-order fractional Yamabe and Hardy-Hénon equations: \((-\Delta)^{\sigma}u=|x|^{-\alpha}u^{p}\quad \text{in}\ \mathbb{R}^{n}\backslash \{0\},\) ( − Δ ) σ u = | x | − α u p in R n ∖ { 0 } , where 0 < σ < n/2, −∞ < α < 2σ, p > 1, and the origin may be a singularity. When the singularity is removable, we establish optimal Liouville-type theorems for all σ ∈ (0, n/2). Moreover, we prove the existence of positive solutions in the supercritical case. For non-removable singularities, we demonstrate the radial symmetry of solutions, which is useful in studying the higher-order fractional singular Yamabe problem.