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2013, 2013(special): 771-780. doi: 10.3934/proc.2013.2013.771

On the uniqueness of blow-up solutions of fully nonlinear elliptic equations

Antonio Vitolo 1, , Maria E. Amendola 1, and  Giulio Galise 1,

1. 

Department of Mathematics, University of Salerno, 84084 Fisciano (SA), Italy, Italy, Italy

Received  September 2012 Revised  December 2012 Published  November 2013

This paper contains new uniqueness results of the boundary blow-up viscosity solutions of second order elliptic equations, generalizing a well known result of Marcus-Veron for the Laplace operator.
Keywords: Elliptic equations, viscosity solutions, blow-up., fully nonlinear equations, maximum principle.
Mathematics Subject Classification: Primary: 35J25, 35B50; Secondary: 35D4.
Citation: Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
References:
[1]

X. Cabré and L. A. Caffarelli, Interior $C^{2,\alpha}$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, J. Math. Pures Appl., 82 (2003), 573-612.  Google Scholar

[2]

L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations'', Colloquium Publications 43, American Mathematical Society, Providence, 1995.  Google Scholar

[3]

L. A. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-398.  Google Scholar

[4]

I. Capuzzo Dolcetta and A. Vitolo, Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst., 28 (2010), 539-557.  Google Scholar

[5]

M. G. Crandall, H.Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  Google Scholar

[6]

M. G. Crandall, M. Kocan, P. L. Lions and A. Swiech, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differ. Equ., 24 (1999), 1-20.  Google Scholar

[7]

M. G. Crandall and A. Swiech, A note on generalized maximum principles for elliptic and parabolic PDE, Lecture Notes in Pure and Appl. Math., 235 (2003), 121-127.  Google Scholar

[8]

F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc. (JEMS), 9 (2007), 317-330.  Google Scholar

[9]

H. Dong, S. Kim and M. Safonov, On uniqueness boundary blow-up solutions of a class of nonlinear elliptic equations, Commun. Partial Differ. Equations, 33 (2008), 177-188.  Google Scholar

[10]

L. Escauriaza, $W^{2,n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  Google Scholar

[11]

M. J. Esteban, P. L. Felmer and A. Quaas, Superlinear elliptic equations for fully nonlinear operators without growth restrictions for the data, Proc. Edinb. Math. Soc., 53 (2010), 125-141.  Google Scholar

[12]

G. Galise and A. Vitolo, Viscosity Solutions of Uniformly Elliptic Equations without Boundary and Growth Conditions at Infinity, Int. J. Differ. Equ., 2011, Article ID 453727, 18 pp.  Google Scholar

[13]

H. Ishii and P. L. Lions, Viscosity Solutions of Fully Nonlinear Second-Order Elliptic Equations, J. Differential Equations, 83 (1990), 26-78.  Google Scholar

[14]

S. Koike, "A Beginners Guide to the Theory of Viscosity Solutions'', MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004.  Google Scholar

[15]

M. Marcus and L. Véron, Uniqueness of solutions with blowup at the boundary for a class of nonlinear elliptic equations, C.R. Acad. Sci. Paris, 317 (1993), 559-563.  Google Scholar

[16]

M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. Henri Poincaré,Analyse non linéaire, 14 (1997), 237-274.  Google Scholar

[17]

M. H. Protter and H. F. Weinberger, "Maximum principles in Differential Equations'', Springer-Verlag, New York, 1984.  Google Scholar

[18]

P. Pucci and J. Serrin, "The maximum principles'', Progress in Nonlinear Differential Equations and Their Applications 73, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[19]

B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607.  Google Scholar

[20]

A. Swiech, $W^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.  Google Scholar

show all references

References:
[1]

X. Cabré and L. A. Caffarelli, Interior $C^{2,\alpha}$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, J. Math. Pures Appl., 82 (2003), 573-612.  Google Scholar

[2]

L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations'', Colloquium Publications 43, American Mathematical Society, Providence, 1995.  Google Scholar

[3]

L. A. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-398.  Google Scholar

[4]

I. Capuzzo Dolcetta and A. Vitolo, Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst., 28 (2010), 539-557.  Google Scholar

[5]

M. G. Crandall, H.Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  Google Scholar

[6]

M. G. Crandall, M. Kocan, P. L. Lions and A. Swiech, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differ. Equ., 24 (1999), 1-20.  Google Scholar

[7]

M. G. Crandall and A. Swiech, A note on generalized maximum principles for elliptic and parabolic PDE, Lecture Notes in Pure and Appl. Math., 235 (2003), 121-127.  Google Scholar

[8]

F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc. (JEMS), 9 (2007), 317-330.  Google Scholar

[9]

H. Dong, S. Kim and M. Safonov, On uniqueness boundary blow-up solutions of a class of nonlinear elliptic equations, Commun. Partial Differ. Equations, 33 (2008), 177-188.  Google Scholar

[10]

L. Escauriaza, $W^{2,n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  Google Scholar

[11]

M. J. Esteban, P. L. Felmer and A. Quaas, Superlinear elliptic equations for fully nonlinear operators without growth restrictions for the data, Proc. Edinb. Math. Soc., 53 (2010), 125-141.  Google Scholar

[12]

G. Galise and A. Vitolo, Viscosity Solutions of Uniformly Elliptic Equations without Boundary and Growth Conditions at Infinity, Int. J. Differ. Equ., 2011, Article ID 453727, 18 pp.  Google Scholar

[13]

H. Ishii and P. L. Lions, Viscosity Solutions of Fully Nonlinear Second-Order Elliptic Equations, J. Differential Equations, 83 (1990), 26-78.  Google Scholar

[14]

S. Koike, "A Beginners Guide to the Theory of Viscosity Solutions'', MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004.  Google Scholar

[15]

M. Marcus and L. Véron, Uniqueness of solutions with blowup at the boundary for a class of nonlinear elliptic equations, C.R. Acad. Sci. Paris, 317 (1993), 559-563.  Google Scholar

[16]

M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. Henri Poincaré,Analyse non linéaire, 14 (1997), 237-274.  Google Scholar

[17]

M. H. Protter and H. F. Weinberger, "Maximum principles in Differential Equations'', Springer-Verlag, New York, 1984.  Google Scholar

[18]

P. Pucci and J. Serrin, "The maximum principles'', Progress in Nonlinear Differential Equations and Their Applications 73, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[19]

B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607.  Google Scholar

[20]

A. Swiech, $W^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.  Google Scholar

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