American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients
  • PROC Home
  • This Issue
  • Next Article
    Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle
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.   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

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.   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

[1]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[2]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[3]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621
[4]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[5]

Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828
[6]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733
[7]

Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465
[8]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[9]

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
[10]

Laurent Di Menza, Olivier Goubet. Stabilizing blow up solutions to nonlinear schrÖdinger equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1059-1082. doi: 10.3934/cpaa.2017051
[11]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027
[12]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[13]

Chuanqiang Chen. On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4761-4811. doi: 10.3934/dcds.2016007
[14]

Chuanqiang Chen. On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3383-3402. doi: 10.3934/dcds.2014.34.3383
[15]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[16]

Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020
[17]

Júlia Matos. Unfocused blow up solutions of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 905-928. doi: 10.3934/dcds.1999.5.905
[18]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure & Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125
[19]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539
[20]

Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435

 Impact Factor: 

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences