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

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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

Abstract
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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. Google Scholar
[2]	L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations'',, Colloquium Publications 43, (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. 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. 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. 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. 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. 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. 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. Google Scholar
[10]	L. Escauriaza, $W^{2,n}$ a priori estimates for solutions to fully nonlinear equations,, Indiana Univ. Math. J., 42 (1993), 413. 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. 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 (4537). Google Scholar
[13]	H. Ishii and P. L. Lions, Viscosity Solutions of Fully Nonlinear Second-Order Elliptic Equations,, J. Differential Equations, 83 (1990), 26. Google Scholar
[14]	S. Koike, "A Beginners Guide to the Theory of Viscosity Solutions'',, MSJ Memoirs 13, (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. 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, 14 (1997), 237. Google Scholar
[17]	M. H. Protter and H. F. Weinberger, "Maximum principles in Differential Equations'',, Springer-Verlag, (1984). Google Scholar
[18]	P. Pucci and J. Serrin, "The maximum principles'',, Progress in Nonlinear Differential Equations and Their Applications 73, (2007). Google Scholar
[19]	B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE,, Arch. Ration. Mech. Anal., 195 (2010), 579. Google Scholar
[20]	A. Swiech, $W^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations,, Adv. Differential Equations, 2 (1997), 1005. Google Scholar

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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. Google Scholar
[2]	L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations'',, Colloquium Publications 43, (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. 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. 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. 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. 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. 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. 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. Google Scholar
[10]	L. Escauriaza, $W^{2,n}$ a priori estimates for solutions to fully nonlinear equations,, Indiana Univ. Math. J., 42 (1993), 413. 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. 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 (4537). Google Scholar
[13]	H. Ishii and P. L. Lions, Viscosity Solutions of Fully Nonlinear Second-Order Elliptic Equations,, J. Differential Equations, 83 (1990), 26. Google Scholar
[14]	S. Koike, "A Beginners Guide to the Theory of Viscosity Solutions'',, MSJ Memoirs 13, (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. 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, 14 (1997), 237. Google Scholar
[17]	M. H. Protter and H. F. Weinberger, "Maximum principles in Differential Equations'',, Springer-Verlag, (1984). Google Scholar
[18]	P. Pucci and J. Serrin, "The maximum principles'',, Progress in Nonlinear Differential Equations and Their Applications 73, (2007). Google Scholar
[19]	B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE,, Arch. Ration. Mech. Anal., 195 (2010), 579. Google Scholar
[20]	A. Swiech, $W^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations,, Adv. Differential Equations, 2 (1997), 1005. Google Scholar

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