Nonexistence of entire positive solutions for conformal Hessian quotient inequalities

Feida Jiang; Xi Chen; Juhua Shi; Feida Jiang; Xi Chen; Juhua Shi

doi:10.3934/era.2021072

Electronic Research Archive

2021, Volume 29, Issue 6: 4075-4086. doi: 10.3934/era.2021072

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Nonexistence of entire positive solutions for conformal Hessian quotient inequalities

Feida Jiang ^{1,2
,
,},
Xi Chen ^{2
,},
Juhua Shi ^{3
,}

1.
School of Mathematics and Shing-Tung Yau Center of Southeast University, Southeast University, Nanjing 211189, China
2.
College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
3.
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China

Received: 01 April 2021 Revised: 01 July 2021 Published: 22 September 2021
Primary: 35J60, 35B08; Secondary: 35B09

In this paper, we consider the nonexistence problem for conformal Hessian quotient inequalities in $ \mathbb{R}^n $. We prove the nonexistence results of entire positive $ k $-admissible solution to a conformal Hessian quotient inequality, and entire $ (k, k') $-admissible solution pair to a system of Hessian quotient inequalities, respectively. We use the contradiction method combining with the integration by parts, suitable choices of test functions, Taylor's expansion and Maclaurin's inequality for Hessian quotient operators.
- Conformal Hessian quotient inequality,
- entire positive solution,
- nonexistence
Citation: Feida Jiang, Xi Chen, Juhua Shi. Nonexistence of entire positive solutions for conformal Hessian quotient inequalities[J]. Electronic Research Archive, 2021, 29(6): 4075-4086. doi: 10.3934/era.2021072

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Abstract

In this paper, we consider the nonexistence problem for conformal Hessian quotient inequalities in $ \mathbb{R}^n $. We prove the nonexistence results of entire positive $ k $-admissible solution to a conformal Hessian quotient inequality, and entire $ (k, k') $-admissible solution pair to a system of Hessian quotient inequalities, respectively. We use the contradiction method combining with the integration by parts, suitable choices of test functions, Taylor's expansion and Maclaurin's inequality for Hessian quotient operators.

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