Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Necessary and sufficient conditions on solvability for Hessian inequalities

Author(s): Xiaohu Ji; Jiguang Bao
Journal: Proc. Amer. Math. Soc. 138 (2010), 175-188.
MSC (2000): Primary 35J60, 35J85
Posted: September 3, 2009
MathSciNet review: 2550182
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: In this paper, we discuss the solvability of the Hessian inequality $\sigma^{\frac{1}{k}}_{k}(\lambda (D^{2}u)) \ge f(u)$ on the entire space $\mathbb{R}^{n}$ and provide a necessary and sufficient condition, which can be regarded as a generalized Keller-Osserman condition.

References:

1.: Haim Brezis, Semilinear equations in $\mathbb{R}^{N}$ without conditions at infinity, Appl. Math. Optim., 12 (1984), 271-282. MR 768633 (86f:35076)
2.: Luis A. Caffarelli, Louis Nirenberg and Joel Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301. MR 806416 (87f:35098)
3.: Sun-Yung Alice Chang, Conformal invariants and partial differential equations, Bull. Amer. Math. Soc. (N.S.), 42 (2005), 365-393. MR 2149088 (2006b:53045)
4.: Kuoshung Cheng and Jenntsann Lin, On the elliptic equations $\Delta u= K(x)u^{\sigma}$ and $\Delta u= K(x)e^{2u}$ , Trans. Amer. Math. Soc., 304 (1987), 639-668. MR 911088 (88j:35054)
5.: Lawrence C. Evans, Classical solutions of fully non-linear, convex, second order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. MR 649348 (83g:35038)
6.: Daniel Faraco and Xiao Zhong, Quasiconvex functions and Hessian equations, Arch. Ration. Mech. Anal., 168 (2003), 245-252. MR 1991516 (2004h:49003)
7.: Marius Ghergu and Vicentiu Radulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal., 17 (2003), 995-1003. MR 2029521 (2004j:34008)
8.: Julian Lopez Gomez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations, 224 (2006), 385-439. MR 2223723 (2007b:35124)
9.: Bo Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. PDE, 8 (1999), 45-69. MR 1666866 (99k:58191)
10.: Philip Hartman, Ordinary differential equations, Second Edition. Birkhäuser, Boston-Basel-Stuttgart, 1982. MR 658490 (83e:34002)
11.: Qinian Jin, Yanyan Li and Haoyuan Xu, Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal., 12 (2005), 441-450. MR 2258318
12.: Jerry L. Kazdan and Frank W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry, 10 (1975), 113-134. MR 0365409 (51:1661)
13.: Joseph B. Keller, On solutions of $\Delta u=f(u)$ , Comm. Pure Appl. Math., 10 (1957), 503-510. MR 0091407 (19:964c)
14.: Nicolai V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk. SSSR Ser. Mat., 46 (1982), 487-523; English transl. in Math. USSR Izv., 20 (1983), 459-492. MR 661144 (84a:35091)
15.: Yuan Lou and Meijun Zhu, Classifications of non-negative solutions to some elliptic problems, Differential Integral Equations, 12 (1999), 601-612. MR 1697247 (2000b:35082)
16.: Jean Mawhin, Duccio Papini and Fabio Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Differential Equations 188 (2003), 33-51. MR 1954507 (2003j:34026)
17.: Robert Osserman, On the inequality $\Delta u\ge f(u),$ Pacific J. Math., 7 (1957), 1641-1647. MR 0098239 (20:4701)
18.: Alessio Porretta and Laurent Veron, Symmetry of large solutions of nonlinear elliptic equations in a ball, J. Functional Analysis, 236 (2006), 581-591. MR 2240175 (2007b:35131)
19.: Neil S. Trudinger and Xujia Wang, Hessian measures. I, Topol. Methods Nonlinear Anal., 10 (1997), 225-239. MR 1634570 (2000a:35061)
20.: Neil S. Trudinger and Xujia Wang, Hessian measures. II, Annals of Mathematics, 150 (1999), 579-604. MR 1726702 (2001f:35141)
21.: Neil S. Trudinger and Xujia Wang, Hessian measures. III, Journal of Functional Analysis, 193 (2002), 1-23. MR 1923626 (2003i:35106)
22.: John Urbas, On the existence of nonclassical solutions for two class of fully nonlinear elliptic equations, Indiana Univ. Math. J., 39 (1990), 355-382. MR 1089043 (92h:35074)
23.: Zuodong Yang, Existence of explosive positive solutions of quasilinear elliptic equations, Appl. Math. Comput., 177 (2006), 581-588. MR 2291983 (2007k:35176)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J60, 35J85

Retrieve articles in all Journals with MSC (2000): 35J60, 35J85

Additional Information:

Xiaohu Ji
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
Email: Ji.Xiaohu@hotmail.com

Jiguang Bao
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
Email: jgbao@bnu.edu.cn

DOI: 10.1090/S0002-9939-09-10032-1
PII: S 0002-9939(09)10032-1
Keywords: Hessian equation, subsolution, existence, nonexistence, Keller-Osserman condition
Received by editor(s): February 20, 2009
Posted: September 3, 2009
Additional Notes: This work was supported by the National Natural Science Foundation of China (10671022) and the Doctoral Programme Foundation of the Institute of Higher Education of China (20060027023).
Communicated by: Matthew J. Gursky
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|