Electronic Research Archive

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

Nonexistence of entire positive solutions for conformal Hessian quotient inequalities

• 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.

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

Related Papers:

• 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.

 [1] T. Aubin, $\acute{E}$quations diff$\acute{e}$rentielles non lin$\acute{e}$aires et probl$\grave{e}$me de Yamabe concernant la courbure scalaire, J. Math. Pur. Appl., 55 (1976), 269–296. (In French.) [2] T. Aubin, Probl$\grave{e}$mes isop$\acute{e}$rim$\acute{e}$triques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573–598. (In French.) [3] Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. (1989) 42: 271-297. [4] The Dirichlet problem for nonlinear second order elliptic equations III: Functions of the eigenvalues of the Hessian. Acta Math. (1985) 155: 261-301. [5] Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. (1981) 34: 525-598. [6] Nonexistence of entire positive solution for a conformal $k$-Hessian inequality. Czechoslovak Math. J. (2020) 70: 311-322. [7] On solutions of $\Delta u = f(u)$. Comm. Pure Appl. Math. (1957) 10: 503-510. [8] On some conformally invariant fully nonlinear equations II: Liouville, Harnack and Yamabe. Acta Math. (2005) 195: 117-154. [9] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing, Singapore, 1996. doi: 10.1142/3302 [10] On the inequality $\Delta u\geq f(u)$. Pacific J. Math. (1957) 7: 1641-1647. [11] Singularities and Liouville theorem for some special conformal Hessian equations. Pav. J. Math. (2013) 266: 117-128. [12] Nonexistence results for Hessian inequality. Methods Appl. Anal. (2010) 17: 213-224. [13] A note on nonexistence of conformal Hessian inequalities. Adv. Math., (China) (2017) 46: 154-158. [14] Conformal deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom. (1984) 20: 479-495. [15] The $k$-Yamabe problem. Surv. Differ. Geom. (2012) 17: 427-457. [16] Remarks concerning the conformal deformamation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (1968) 22: 265-274. [17] On the Dirichlet problem for Hessian equations. Acta Math. (1995) 175: 151-164. [18] Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. (2000) 101: 283-316. [19] On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. (1960) 12: 21-37.
• Reader Comments
• © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.604 0.8

Metrics

Article views(567) PDF downloads(104) Cited by(0)

Article outline

Other Articles By Authors

• On This Site
• On Google Scholar

/

DownLoad:  Full-Size Img  PowerPoint