Entire subsolutions of Monge-Ampère type equations

Limei Dai; Hongyu Li

doi:10.3934/cpaa.2020002

Article Contents

2020, Volume 19, Issue 1: 19-30. Doi: 10.3934/cpaa.2020002

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

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
^* Corresponding author

Received: January 31, 2018

Revised: February 28, 2019

Published: July 2019

The first author is supported by Shandong Provincial Natural Science Foundation (ZR2018LA006)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35J96, 35B08; Secondary: 35A01.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.