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

Entire subsolutions of Monge-Ampère type equations

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.

Citation: Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure and 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.

[2]

J. G. BaoX. 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.

[3]

L. A. CaffarelliL. 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.

[4]

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

[5]

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

[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.

[7]

Y. HuangF. 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.

[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.

[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.

[10]

F. D. JiangN. 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.

[11]

Q. N. JinY. 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.

[12]

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

[13]

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

[14]

J. K. LiuN. 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.

[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.

[16]

X. N. MaN. 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.

[17]

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

[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.

[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. 

[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.

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.

[2]

J. G. BaoX. 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.

[3]

L. A. CaffarelliL. 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.

[4]

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

[5]

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

[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.

[7]

Y. HuangF. 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.

[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.

[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.

[10]

F. D. JiangN. 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.

[11]

Q. N. JinY. 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.

[12]

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

[13]

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

[14]

J. K. LiuN. 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.

[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.

[16]

X. N. MaN. 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.

[17]

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

[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.

[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. 

[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.

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