American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Infinitely many solutions and Morse index for non-autonomous elliptic problems
  • CPAA Home
  • This Issue
  • Next Article
    Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats
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

[1]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121
[2]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991
[3]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59
[4]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825
[5]

Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060
[6]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
[7]

Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061
[8]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347
[9]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705
[10]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559
[11]

Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations & Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015
[12]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221
[13]

Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058
[14]

Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053
[15]

Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447
[16]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697
[17]

Limei Dai. Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 579-596. doi: 10.3934/dcds.2020023
[18]

Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607
[19]

Peter Poláčik, Darío A. Valdebenito. Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1369-1393. doi: 10.3934/dcdss.2020077
[20]

Tomasz Komorowski, Adam Bobrowski. A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020248

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (95)
  • HTML views (138)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences