Necessary and sufficient conditions on solvability for Hessian inequalities

Ji, Xiaohu; Bao, Jiguang

doi:10.1090/S0002-9939-09-10032-1

Necessary and sufficient conditions on solvability for Hessian inequalities
HTML articles powered by AMS MathViewer

by Xiaohu Ji and Jiguang Bao PDF

Proc. Amer. Math. Soc. 138 (2010), 175-188 Request permission

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

H. Brezis, Semilinear equations in $\textbf {R}^N$ without condition at infinity, Appl. Math. Optim. 12 (1984), no. 3, 271–282. MR 768633, DOI 10.1007/BF01449045
L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), no. 3-4, 261–301. MR 806416, DOI 10.1007/BF02392544
Sun-Yung Alice Chang, Conformal invariants and partial differential equations, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 3, 365–393. MR 2149088, DOI 10.1090/S0273-0979-05-01058-X
Kuo-Shung Cheng and Jenn-Tsann Lin, On the elliptic equations $\Delta u=K(x)u^\sigma$ and $\Delta u=K(x)e^{2u}$, Trans. Amer. Math. Soc. 304 (1987), no. 2, 639–668. MR 911088, DOI 10.1090/S0002-9947-1987-0911088-1
Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303
Daniel Faraco and Xiao Zhong, Quasiconvex functions and Hessian equations, Arch. Ration. Mech. Anal. 168 (2003), no. 3, 245–252. MR 1991516, DOI 10.1007/s00205-003-0255-8
Marius Ghergu and Vicenţiu Rădulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal. 17 (2003), 995–1003. MR 2029521, DOI 10.1155/S1085337503305020
Julián López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006), no. 2, 385–439. MR 2223723, DOI 10.1016/j.jde.2005.08.008
Bo Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations 8 (1999), no. 1, 45–69. MR 1666866, DOI 10.1007/s005260050116
Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR 658490
Qinian Jin, Yanyan Li, and Haoyuan Xu, Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal. 12 (2005), no. 4, 441–449. MR 2258318, DOI 10.4310/MAA.2005.v12.n4.a5
Jerry L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113–134. MR 365409
J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math. 10 (1957), 503–510. MR 91407, DOI 10.1002/cpa.3160100402
N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670 (Russian). MR 661144
Yuan Lou and Meijun Zhu, Classifications of nonnegative solutions to some elliptic problems, Differential Integral Equations 12 (1999), no. 4, 601–612. MR 1697247
Jean Mawhin, Duccio Papini, and Fabio Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Differential Equations 188 (2003), no. 1, 33–51. MR 1954507, DOI 10.1016/S0022-0396(02)00073-6
Robert Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math. 7 (1957), 1641–1647. MR 98239
Alessio Porretta and Laurent Véron, Symmetry of large solutions of nonlinear elliptic equations in a ball, J. Funct. Anal. 236 (2006), no. 2, 581–591. MR 2240175, DOI 10.1016/j.jfa.2006.03.010
Neil S. Trudinger and Xu-Jia Wang, Hessian measures. I, Topol. Methods Nonlinear Anal. 10 (1997), no. 2, 225–239. Dedicated to Olga Ladyzhenskaya. MR 1634570, DOI 10.12775/TMNA.1997.030
Neil S. Trudinger and Xu-Jia Wang, Hessian measures. II, Ann. of Math. (2) 150 (1999), no. 2, 579–604. MR 1726702, DOI 10.2307/121089
Neil S. Trudinger and Xu-Jia Wang, Hessian measures. III, J. Funct. Anal. 193 (2002), no. 1, 1–23. MR 1923626, DOI 10.1006/jfan.2001.3925
John I. E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J. 39 (1990), no. 2, 355–382. MR 1089043, DOI 10.1512/iumj.1990.39.39020
Zuodong Yang, Existence of explosive positive solutions of quasilinear elliptic equations, Appl. Math. Comput. 177 (2006), no. 2, 581–588. MR 2291983, DOI 10.1016/j.amc.2005.09.088