doi: 10.3934/era.2021072
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Nonexistence of entire positive solutions for conformal Hessian quotient inequalities

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

* Corresponding author: Feida Jiang

Received  April 2021 Revised  July 2021 Early access September 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11771214 and No.11971157)

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. Electronic Research Archive, doi: 10.3934/era.2021072
References:
[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.)  Google Scholar

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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.)  Google Scholar

[3]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations III: Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.  Google Scholar

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B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[6]

F. JiangS. Cui and G. Li, Nonexistence of entire positive solution for a conformal $k$-Hessian inequality, Czechoslovak Math. J., 70 (2020), 311-322.  doi: 10.21136/CMJ.2019.0289-18.  Google Scholar

[7]

J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.  Google Scholar

[8]

A. Li and Y. Y. Li, On some conformally invariant fully nonlinear equations II: Liouville, Harnack and Yamabe, Acta Math., 195 (2005), 117-154.  doi: 10.1007/BF02588052.  Google Scholar

[9]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing, Singapore, 1996. doi: 10.1142/3302.  Google Scholar

[10]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.   Google Scholar

[11]

Q. Ou, Singularities and Liouville theorem for some special conformal Hessian equations, Pav. J. Math., 266 (2013), 117-128.  doi: 10.2140/pjm.2013.266.117.  Google Scholar

[12]

Q. Ou, Nonexistence results for Hessian inequality, Methods Appl. Anal., 17 (2010), 213-224.  doi: 10.4310/MAA.2010.v17.n2.a5.  Google Scholar

[13]

Q. Ou, A note on nonexistence of conformal Hessian inequalities, Adv. Math., (China), 46 (2017), 154-158.   Google Scholar

[14]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom., 20 (1984), 479-495.   Google Scholar

[15]

W. ShengN. S. Trudinger and X.-J. Wang, The $k$-Yamabe problem, Surv. Differ. Geom., 17 (2012), 427-457.  doi: 10.4310/SDG.2012.v17.n1.a10.  Google Scholar

[16]

N. S. Trudinger, Remarks concerning the conformal deformamation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 265-274.   Google Scholar

[17]

N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303.  Google Scholar

[18]

J. A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., 101 (2000), 283-316.  doi: 10.1215/S0012-7094-00-10127-5.  Google Scholar

[19]

H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 12 (1960), 21-37.   Google Scholar

show all references

References:
[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.)  Google Scholar

[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.)  Google Scholar

[3]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations III: Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.  Google Scholar

[5]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[6]

F. JiangS. Cui and G. Li, Nonexistence of entire positive solution for a conformal $k$-Hessian inequality, Czechoslovak Math. J., 70 (2020), 311-322.  doi: 10.21136/CMJ.2019.0289-18.  Google Scholar

[7]

J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.  Google Scholar

[8]

A. Li and Y. Y. Li, On some conformally invariant fully nonlinear equations II: Liouville, Harnack and Yamabe, Acta Math., 195 (2005), 117-154.  doi: 10.1007/BF02588052.  Google Scholar

[9]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing, Singapore, 1996. doi: 10.1142/3302.  Google Scholar

[10]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.   Google Scholar

[11]

Q. Ou, Singularities and Liouville theorem for some special conformal Hessian equations, Pav. J. Math., 266 (2013), 117-128.  doi: 10.2140/pjm.2013.266.117.  Google Scholar

[12]

Q. Ou, Nonexistence results for Hessian inequality, Methods Appl. Anal., 17 (2010), 213-224.  doi: 10.4310/MAA.2010.v17.n2.a5.  Google Scholar

[13]

Q. Ou, A note on nonexistence of conformal Hessian inequalities, Adv. Math., (China), 46 (2017), 154-158.   Google Scholar

[14]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom., 20 (1984), 479-495.   Google Scholar

[15]

W. ShengN. S. Trudinger and X.-J. Wang, The $k$-Yamabe problem, Surv. Differ. Geom., 17 (2012), 427-457.  doi: 10.4310/SDG.2012.v17.n1.a10.  Google Scholar

[16]

N. S. Trudinger, Remarks concerning the conformal deformamation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 265-274.   Google Scholar

[17]

N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303.  Google Scholar

[18]

J. A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., 101 (2000), 283-316.  doi: 10.1215/S0012-7094-00-10127-5.  Google Scholar

[19]

H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 12 (1960), 21-37.   Google Scholar

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