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January  2020, 40(1): 579-596. doi: 10.3934/dcds.2020023

Existence and nonexistence of subsolutions for augmented Hessian equations

Limei Dai

School of Mathematics and Information Science, Weifang University, Weifang 261061, China

Received  April 2019 Published  October 2019

Fund Project: The author is supported by Shandong Provincial Natural Science Foundation, China (ZR2018LA006).

In this paper, we consider the augmented Hessian equations $ S_k^{\frac{1}{k}}[D^2u+\sigma(x)I] = f(u) $ in $ \mathbb{R}^{n} $ or $ \mathbb{R}^{n}_+ $. We first give the necessary and sufficient condition of the existence of classical subsolutions to the equations in $ \mathbb{R}^{n} $ for $ \sigma(x) = \alpha $, which is an extended Keller-Osserman condition. Then we obtain the nonexistence of positive viscosity subsolutions of the equations in $ \mathbb{R}^{n} $ or $ \mathbb{R}^{n}_+ $ for $ f(u) = u^p $ with $ p>1 $.

Keywords: Augmented Hessian equations, subsolutions, Keller-Osserman condition.
Mathematics Subject Classification: 35J60, 35D40.
Citation: Limei Dai. Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 579-596. doi: 10.3934/dcds.2020023
References:
[1]

J. G. Bao, X. H. Ji and H. G. Li, Existence and nonexistence theorem for entire subsolutions of $k$-Yamabe type equations, J. Differential Equations, 253 (2012), 2140-2160.  doi: 10.1016/j.jde.2012.06.018.  Google Scholar

[2]

L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, Ⅰ. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.  Google Scholar

[3]

L. Caffarelli, L. 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

[4]

I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, Entire subsolutions of fully nonlinear degenerate elliptic equations, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 147-161.   Google Scholar

[5]

I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, On the inequality $F(x, D^2u)\geq f(u)+g(u)|Du|^q$, Math. Ann., 365 (2016), 423-448.  doi: 10.1007/s00208-015-1280-2.  Google Scholar

[6]

H. Car and R. Pröpper, Removable singularities of $m$-Hessian equations, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 6, 18 pp. doi: 10.1007/s00030-016-0429-3.  Google Scholar

[7]

K. S. Chou and X. J. Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math., 54 (2001), 1029-1064.  doi: 10.1002/cpa.1016.  Google Scholar

[8]

D. P. Covei, The Keller-Osserman problem for the $k$-Hessian operator, arXiv: 1508.04653. Google Scholar

[9]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 219-245.  doi: 10.1016/S0294-1449(00)00109-8.  Google Scholar

[11]

S. Dumont, L. Dupaigne, O. Goubet and V. Radulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.  doi: 10.1515/ans-2007-0205.  Google Scholar

[12]

P. L. Felmer and A. Quaas, On critical exponents for the Pucci's extremal operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 843-865.  doi: 10.1016/S0294-1449(03)00011-8.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

B. Guan, Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.  doi: 10.1215/00127094-2713591.  Google Scholar

[15]

Y. Huang, F. D. Jiang and J. K. Liu, Boundary $C^ {2, \alpha}$ estimates for Monge-Ampère type equations, Adv. Math., 281 (2015), 706-733.  doi: 10.1016/j.aim.2014.12.043.  Google Scholar

[16]

X. H. Ji and J. G. Bao, Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc., 138 (2010), 175-188.  doi: 10.1090/S0002-9939-09-10032-1.  Google Scholar

[17]

F. D. Jiang and N. S. Trudinger, On the Dirichlet problem for general augmented Hessian equations, arXiv: 1903.12410. Google Scholar

[18]

F. D. Jiang, N. S. Trudinger and X. P. Yang, On the Dirichlet problem for Monge-Ampère type equations, Calc. Var. Partial Differential Equations, 49 (2014), 1223-1236.  doi: 10.1007/s00526-013-0619-3.  Google Scholar

[19]

F. D. Jiang, N. S. Trudinger and X. P. Yang, On the Dirichlet problem for a class of augmented Hessian equations, J. Differential Equations, 258 (2015), 1548-1576.  doi: 10.1016/j.jde.2014.11.005.  Google Scholar

[20]

Q. N. Jin, Y. Y. Li and H. Y. Xu, Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal., 12 (2005), 441-449.  doi: 10.4310/MAA.2005.v12.n4.a5.  Google Scholar

[21]

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

[22]

Y. Y. Li, Some existence results for fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.  Google Scholar

[23]

J. K. Liu, N. S. Trudinger and X. J. Wang, Interior $C^ {2, \alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184.  doi: 10.1080/03605300903236609.  Google Scholar

[24]

G. Z. Lu and J. Y. Zhu, The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differential Equations, 258 (2015), 2054-2079.  doi: 10.1016/j.jde.2014.11.022.  Google Scholar

[25]

X. N. Ma, N. S. Trudinger and X. J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.  doi: 10.1007/s00205-005-0362-9.  Google Scholar

[26]

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

[27]

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

[28]

X. J. Wang, The $k$-Hessian equation, Geometric analysis and PDEs, 177–252, Lecture Notes in Math., 1977, Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-01674-5_5.  Google Scholar

show all references

References:
[1]

J. G. Bao, X. H. Ji and H. G. Li, Existence and nonexistence theorem for entire subsolutions of $k$-Yamabe type equations, J. Differential Equations, 253 (2012), 2140-2160.  doi: 10.1016/j.jde.2012.06.018.  Google Scholar

[2]

L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, Ⅰ. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.  Google Scholar

[3]

L. Caffarelli, L. 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

[4]

I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, Entire subsolutions of fully nonlinear degenerate elliptic equations, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 147-161.   Google Scholar

[5]

I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, On the inequality $F(x, D^2u)\geq f(u)+g(u)|Du|^q$, Math. Ann., 365 (2016), 423-448.  doi: 10.1007/s00208-015-1280-2.  Google Scholar

[6]

H. Car and R. Pröpper, Removable singularities of $m$-Hessian equations, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 6, 18 pp. doi: 10.1007/s00030-016-0429-3.  Google Scholar

[7]

K. S. Chou and X. J. Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math., 54 (2001), 1029-1064.  doi: 10.1002/cpa.1016.  Google Scholar

[8]

D. P. Covei, The Keller-Osserman problem for the $k$-Hessian operator, arXiv: 1508.04653. Google Scholar

[9]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 219-245.  doi: 10.1016/S0294-1449(00)00109-8.  Google Scholar

[11]

S. Dumont, L. Dupaigne, O. Goubet and V. Radulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.  doi: 10.1515/ans-2007-0205.  Google Scholar

[12]

P. L. Felmer and A. Quaas, On critical exponents for the Pucci's extremal operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 843-865.  doi: 10.1016/S0294-1449(03)00011-8.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

B. Guan, Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.  doi: 10.1215/00127094-2713591.  Google Scholar

[15]

Y. Huang, F. D. Jiang and J. K. Liu, Boundary $C^ {2, \alpha}$ estimates for Monge-Ampère type equations, Adv. Math., 281 (2015), 706-733.  doi: 10.1016/j.aim.2014.12.043.  Google Scholar

[16]

X. H. Ji and J. G. Bao, Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc., 138 (2010), 175-188.  doi: 10.1090/S0002-9939-09-10032-1.  Google Scholar

[17]

F. D. Jiang and N. S. Trudinger, On the Dirichlet problem for general augmented Hessian equations, arXiv: 1903.12410. Google Scholar

[18]

F. D. Jiang, N. S. Trudinger and X. P. Yang, On the Dirichlet problem for Monge-Ampère type equations, Calc. Var. Partial Differential Equations, 49 (2014), 1223-1236.  doi: 10.1007/s00526-013-0619-3.  Google Scholar

[19]

F. D. Jiang, N. S. Trudinger and X. P. Yang, On the Dirichlet problem for a class of augmented Hessian equations, J. Differential Equations, 258 (2015), 1548-1576.  doi: 10.1016/j.jde.2014.11.005.  Google Scholar

[20]

Q. N. Jin, Y. Y. Li and H. Y. Xu, Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal., 12 (2005), 441-449.  doi: 10.4310/MAA.2005.v12.n4.a5.  Google Scholar

[21]

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

[22]

Y. Y. Li, Some existence results for fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.  Google Scholar

[23]

J. K. Liu, N. S. Trudinger and X. J. Wang, Interior $C^ {2, \alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184.  doi: 10.1080/03605300903236609.  Google Scholar

[24]

G. Z. Lu and J. Y. Zhu, The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differential Equations, 258 (2015), 2054-2079.  doi: 10.1016/j.jde.2014.11.022.  Google Scholar

[25]

X. N. Ma, N. S. Trudinger and X. J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.  doi: 10.1007/s00205-005-0362-9.  Google Scholar

[26]

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

[27]

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

[28]

X. J. Wang, The $k$-Hessian equation, Geometric analysis and PDEs, 177–252, Lecture Notes in Math., 1977, Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-01674-5_5.  Google Scholar

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