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January  2020, 19(1): 19-30. doi: 10.3934/cpaa.2020002

Entire subsolutions of Monge-Ampère type equations

Limei Dai 1,, and  Hongyu Li 2,

1. 

School of Mathematics and Information Science, Weifang University, Weifang, 261061, China
2. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

* Corresponding author

Received  February 2018 Revised  March 2019 Published  July 2019

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

In this paper, we consider the subsolutions of the Monge-Ampère type equations $ {\det}^{\frac{1}{n}}(D^2u+\alpha I) = f(u) $ in $ \mathbb{R}^{n} $. We obtain the necessary and sufficient condition of the existence of subsolutions.

Keywords: Monge-Ampère type equations, entire subsolutions, existence.
Mathematics Subject Classification: Primary: 35J96, 35B08; Secondary: 35A01.
Citation: Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 19-30. doi: 10.3934/cpaa.2020002
References:
[1]

H. Brezis, Semilinear equations in ${\mathbb R}^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.  doi: 10.1007/BF01449045.  Google Scholar

[2]

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

[3]

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

[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]

P. F. Guan and X. J. Wang, On a Monge-Ampère equation arising in geometric optics, J. Differential Geom., 48 (1998), 205-223.   Google Scholar

[6]

Q. Han and J. X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, 130, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/130.  Google Scholar

[7]

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

[8]

N. M. Ivochkina, Classical solvability of the Dirichlet problem for the Monge-Ampère equation, (Russian), Questions in Quantum Field Theory and Statistical Physics, 4. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)), 131 (1983), 72–79.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.   Google Scholar

[14]

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

[15]

C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, (1974), 245–272.  Google Scholar

[16]

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

[17]

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

[18]

N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, Adv. Lect. Math. (Handbook of geometric analysis, a collection of papers dedicated to Lipman Bers), Int. Press, Somerville, MA, 1 (2008), 467–524.  Google Scholar

[19]

N. S. Trudinger and X. J. Wang, On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 143-174.   Google Scholar

[20]

X. J. Wang, On the design of a reflector antenna, Inverse Problems, 12 (1996), 351-375.  doi: 10.1088/0266-5611/12/3/013.  Google Scholar

show all references

References:
[1]

H. Brezis, Semilinear equations in ${\mathbb R}^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.  doi: 10.1007/BF01449045.  Google Scholar

[2]

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

[3]

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

[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]

P. F. Guan and X. J. Wang, On a Monge-Ampère equation arising in geometric optics, J. Differential Geom., 48 (1998), 205-223.   Google Scholar

[6]

Q. Han and J. X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, 130, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/130.  Google Scholar

[7]

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

[8]

N. M. Ivochkina, Classical solvability of the Dirichlet problem for the Monge-Ampère equation, (Russian), Questions in Quantum Field Theory and Statistical Physics, 4. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)), 131 (1983), 72–79.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.   Google Scholar

[14]

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

[15]

C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, (1974), 245–272.  Google Scholar

[16]

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

[17]

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

[18]

N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, Adv. Lect. Math. (Handbook of geometric analysis, a collection of papers dedicated to Lipman Bers), Int. Press, Somerville, MA, 1 (2008), 467–524.  Google Scholar

[19]

N. S. Trudinger and X. J. Wang, On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 143-174.   Google Scholar

[20]

X. J. Wang, On the design of a reflector antenna, Inverse Problems, 12 (1996), 351-375.  doi: 10.1088/0266-5611/12/3/013.  Google Scholar

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