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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

 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.
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. [2] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations'', Colloquium Publications 43, American Mathematical Society, Providence, 1995. [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. [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. [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. [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. [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. [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. [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. [10] L. Escauriaza, $W^{2,n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423. [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. [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. [13] H. Ishii and P. L. Lions, Viscosity Solutions of Fully Nonlinear Second-Order Elliptic Equations, J. Differential Equations, 83 (1990), 26-78. [14] S. Koike, "A Beginners Guide to the Theory of Viscosity Solutions'', MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004. [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. [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. [17] M. H. Protter and H. F. Weinberger, "Maximum principles in Differential Equations'', Springer-Verlag, New York, 1984. [18] P. Pucci and J. Serrin, "The maximum principles'', Progress in Nonlinear Differential Equations and Their Applications 73, Birkhäuser Verlag, Basel, 2007. [19] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607. [20] A. Swiech, $W^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.

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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), 573-612. [2] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations'', Colloquium Publications 43, American Mathematical Society, Providence, 1995. [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. [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. [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. [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. [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. [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. [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. [10] L. Escauriaza, $W^{2,n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423. [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. [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. [13] H. Ishii and P. L. Lions, Viscosity Solutions of Fully Nonlinear Second-Order Elliptic Equations, J. Differential Equations, 83 (1990), 26-78. [14] S. Koike, "A Beginners Guide to the Theory of Viscosity Solutions'', MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004. [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. [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. [17] M. H. Protter and H. F. Weinberger, "Maximum principles in Differential Equations'', Springer-Verlag, New York, 1984. [18] P. Pucci and J. Serrin, "The maximum principles'', Progress in Nonlinear Differential Equations and Their Applications 73, Birkhäuser Verlag, Basel, 2007. [19] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607. [20] A. Swiech, $W^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.
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