Entire subsolutions of Monge-Ampère type equations

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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 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. 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.
[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.
[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.
[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. 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.
[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. 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.
[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.
[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. 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.
[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. 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.
[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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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. 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.
[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.
[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.
[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. 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.
[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. 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.
[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.
[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. 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.
[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. 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.
[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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