In this paper, we prove some properties of Hessian operators of asymmetric matrix, and study Liouville type Theorems of p-Hessian type equations. By integration by parts, we prove the nonexistence results of positive solutions of p-k-Hessian equations and inequalities with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3277_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$p \in (1, n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3277_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le k < \frac{n}{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo><</mo> <mfrac> <mi>n</mi> <mi>p</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Moreover, by Newton–Maclaurin inequality, we prove some nonexistence results of positive solutions of p-k-Hessian equations and inequalities with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3277_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$p \in (1, n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3277_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{n}{p} \le k \le n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mi>n</mi> <mi>p</mi> </mfrac> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, p-Hessian quotient equations and inequalities and mixed p-Hessian quotient equations and inequalities.

Nonexistence results of positive solutions of p-Hessian type equations and inequalities

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发表日期：

2025年9月3日

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Chuanqiang Chen,
Lianggen Hu,
Botao Wang

摘要

In this paper, we prove some properties of Hessian operators of asymmetric matrix, and study Liouville type Theorems of p-Hessian type equations. By integration by parts, we prove the nonexistence results of positive solutions of p-k-Hessian equations and inequalities with $p \in (1, n)$ $p \in (1, n)$ and $1 \le k < \frac{n}{p}$ $1 \leq k < \frac{n}{p}$ . Moreover, by Newton–Maclaurin inequality, we prove some nonexistence results of positive solutions of p-k-Hessian equations and inequalities with $p \in (1, n)$ $p \in (1, n)$ and $\frac{n}{p} \le k \le n$ $\frac{n}{p} \leq k \leq n$ , p-Hessian quotient equations and inequalities and mixed p-Hessian quotient equations and inequalities.