# American Institute of Mathematical Sciences

November  2011, 10(6): 1707-1714. doi: 10.3934/cpaa.2011.10.1707

## Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations

 1 School of Mathematics and Information Science, Weifang University, Weifang 261061, China

Received  June 2010 Revised  January 2011 Published  May 2011

In this paper, we use the Perron method to prove the existence of viscosity solutions with asymptotic behavior at infinity to fully nonlinear uniformly elliptic equations in $R^n$.
Citation: Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
##### References:
 [1] J. G. Bao, Fully nonlinear elliptic equations on general domains, Canad. J. Math., 54 (2002), 1121-1141. doi: 10.4153/CJM-2002-042-9.  Google Scholar [2] Luis A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.  Google Scholar [3] X. Cabré and Luis A. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations $F(D^2u)=0$, Topol. Methods Nonlinear Anal., 6 (1995), 31-48.  Google Scholar [4] L. Caffarelli and Y. Y. Li, An extension to a theorem of Jörgens, Calabi, and Pogorelov, Comm. Pure Appl. Math., 56 (2003), 549-583. doi: 10.1002/cpa.10067.  Google Scholar [5] L. M. Dai and J. G. Bao, Entire solutions with asymptotic behavior of Hessian equations,, Adv. Math. (China), ().   Google Scholar [6] Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. doi: 10.1002/cpa.3160350303.  Google Scholar [7] D. Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, Berlin, 1983.  Google Scholar [8] H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math., 42 (1989), 15-45. doi: 10.1002/cpa.3160420103.  Google Scholar [9] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain. (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.  Google Scholar [10] O. Savin, Entire solutions to a class of fully nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 369-405.  Google Scholar [11] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607. doi: 10.1007/s00205-009-0218-9.  Google Scholar

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##### References:
 [1] J. G. Bao, Fully nonlinear elliptic equations on general domains, Canad. J. Math., 54 (2002), 1121-1141. doi: 10.4153/CJM-2002-042-9.  Google Scholar [2] Luis A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.  Google Scholar [3] X. Cabré and Luis A. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations $F(D^2u)=0$, Topol. Methods Nonlinear Anal., 6 (1995), 31-48.  Google Scholar [4] L. Caffarelli and Y. Y. Li, An extension to a theorem of Jörgens, Calabi, and Pogorelov, Comm. Pure Appl. Math., 56 (2003), 549-583. doi: 10.1002/cpa.10067.  Google Scholar [5] L. M. Dai and J. G. Bao, Entire solutions with asymptotic behavior of Hessian equations,, Adv. Math. (China), ().   Google Scholar [6] Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. doi: 10.1002/cpa.3160350303.  Google Scholar [7] D. Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, Berlin, 1983.  Google Scholar [8] H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math., 42 (1989), 15-45. doi: 10.1002/cpa.3160420103.  Google Scholar [9] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain. (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.  Google Scholar [10] O. Savin, Entire solutions to a class of fully nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 369-405.  Google Scholar [11] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607. doi: 10.1007/s00205-009-0218-9.  Google Scholar
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