2013, 12(2): 621-645. doi: 10.3934/cpaa.2013.12.621

Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations

1. 

AGM UMR 8088, Universié de Cergy Pontoise, UFR Sciences et techniques Site Saint Martin, BP 222 95302, Cergy Pontoise Cedex, France

2. 

LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex

Received  July 2011 Revised  February 2012 Published  September 2012

We prove here the existence of boundary blow up solutions for fully nonlinear equations in general domains, for a nonlinearity satisfying Keller-Osserman type condition. If moreover the nonlinearity is non decreasing, we prove uniqueness for boundary blow up solutions on balls for operators related to Pucci's operators.
Citation: 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
References:
[1]

C. Bandle and M. Essen, On the solutions of quasilinear elliptic problems with boundary blow-up,, in, (1994), 93.

[2]

C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations, existence, uniqueness and asymptotic behaviour,, Journal d'analyse math\'ematique, 58 (1992), 9. doi: 10.1007/BF02790355.

[3]

C. Bandle and M. Marcus, Asymptotic behaviour of solutions and their derivatives for semilinear elliptic problems with blowup on the boundary,, Ann. Inst. Henri Poincar\'e, 12 (1995), 155.

[4]

C. Bandle and H. Leutwiler, On a quasilinear elliptic equation and a Riemannian metric invariant under Mobius transformation,, Aequationes Math., 42 (1991), 166. doi: 10.1007/BF01818488.

[5]

C. Bandle and S. Vernier-Piro, Estimates for solutions of quasilinear problems with dead cores,, Za Math. Phys, 54 (2003), 815. doi: 10.1007/s00033-003-3203-4.

[6]

L. Bieberbach, $\Delta u = e^u$ und die automorphen funktionen,, Math. Ann., 77 (1916), 173. doi: 10.1007/BF01456901.

[7]

I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators,, Ann. Fac. Sci Toulouse Math, 13 (2004), 261. doi: 10.5802/afst.1070.

[8]

I. Birindelli and F. Demengel, First eigenvalue and Maximum principle for fully nonlinear singular operators,, Advances in Differential Equations, 11 (2006), 91.

[9]

I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators,, Comm. Pure and Appl. Analysis, 6 (2007), 335. doi: 10.3934/cpaa.2007.6.335.

[10]

I. Birindelli and F. Demengel, The Dirichlet problem for singular fully nonlinear operators,, Discrete and Continuous Dynamical Sytems, (2007), 110.

[11]

I. Birindelli and F. Demengel, Regularity of radial solutions for degenerate fully nonlinear equations,, Arxiv 0339276, (0339).

[12]

I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully non linear operators,, Electronic Journal of Differential Equations, 249 (2010), 1089. doi: 10.1016/j.jde.2010.03.015.

[13]

I. Birindelli and F. Demengel, Uniqueness of the first eigenfunction for fully nonlinear equations: the radial case,, Journal for analysis and its applications, 29 (2010), 75. doi: 10.4171/ZAA/1398.

[14]

I. Birindelli and F. Demengel, Eigenfunctions for singular fully nonlinear equations in unbounded domains,, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 697. doi: 10.1007/s00030-010-0077-y.

[15]

L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations,, Ann. Math., 130 (1989), 189.

[16]

X. Cabre and L. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations F(D2u) = 0,, Topological Meth. Nonlinear Anal., 6 (1995), 31.

[17]

X. Cabre and L. Caffarelli, Interior $C^2$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, , J. Maths Pures Appl., 82 (2003), 573. doi: 10.1016/S0021-7824(03)00029-1.

[18]

I. Capuzzo Dolcetta and A. Vitolo, Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations,, Discrete and Continuous Dynamical Systems (DCDS-A), 28 (2010), 539. doi: 10.3934/dcds.2010.28.539.

[19]

O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry,, J. Differential Equations, 249 (2010), 931. doi: 10.1016/j.jde.2010.02.023.

[20]

O. Costin, L. Dupaigne and O. Goubet, Uniqueness of large solutions,, J. Math. Analysis, 395 (2012), 806. doi: 10.1016/j.jmaa.2012.05.085.

[21]

D. P. Covei, Existence of solutions to quasilinear elliptic problems with boundary blow up,, An. Univ. Oradea Fasc. Mat., 17 (2010).

[22]

G. Davila, P. Felmer and A. Quaas, Harnack Inequality for singular fully nonlinear operators and some existence's results,, Calculus of Variations and PDE, 39 (2010), 557. doi: 10.1007/s00526-010-0325-3.

[23]

M. Del Pino and R. Manasevich, Global bifurcation from the eigenvalues of the p-Laplacian,, Journal of Differential Equations, 92 (1991), 226. doi: 10.1016/0022-0396(91)90048-E.

[24]

G. Diaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness,, Nonlinear Analysis. Theory, 20 (1993), 97. doi: 10.1016/0362-546X(93)90012-H.

[25]

S. Dumont, L. Dupaigne, O. Goubet and V. Radulescu, Back to the Keller-Osserman condition for boundary blow up solutions,, Advanced Nonlinear Studies, 7 (2007), 271.

[26]

M. Esteban, P. Felmer and A. Quaas, Super-linear elliptic equation for fully nonlinear operators without growth restrictions for the data,, Proc. Roy. Soc. Edinburgh, 53 (2010), 125. doi: 10.1017/S0013091507001393.

[27]

C. Imbert, Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate fully non-linear elliptic equations,, Journal of Differential Equations, 250 (2011), 1553. doi: 10.1016/j.jde.2010.07.005.

[28]

J. B. Keller, On solutions of $\Delta u =f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503. doi: 10.1002/cpa.3160100402.

[29]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary value problem,, Proc. Amer. Math. Soc., 111 (1991), 721. doi: 10.1090/S0002-9939-1991-1037213-9.

[30]

C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations,, in, (1974).

[31]

M. Marcus and L. Véron, Uniqueness and asymptotic behaviour of solutions with boundary blowup for a class of nonlinear elliptic equations,, Ann. Inst. Henri Poincar\'e, 14 (1997), 237. doi: 10.1016/S0294-1449(97)80146-1.

[32]

M. Marcus and L. Véron, Existence and uniqueness results for large solutions of nonlinear elliptic equations,, J. Evol. Equations, 3 (2004), 637.

[33]

J. Matero, Quasilinear elliptic equations with boundary blow up,, Journal d'analyse math\'ematique, 69 (1996), 229. doi: 10.1007/BF02787108.

[34]

N. Nadirashvili and S. Vladut, On axially symmetric solutions of fully nonlinear elliptic equations,, To appear in Math. Z., ().

[35]

P. Pucci and J. Serrin, Dead Cores and Bursts for quasilinear Singular elliptic equations,, SIAM Journal of Math. Analysis, 38 (2006), 259. doi: 10.1137/050630027.

[36]

R. Osserman, On the inequality $\Delta u \geq f(u)$,, Pacific J. Math., 7 (1957), 1641.

[37]

S. L. Pohozaev, The Dirichlet problem for the equation $\Delta u = u^2$,, Sov. Math. Dokl., 1 (1961), 1143.

[38]

H. Rademacher, Einige besondere Probleme der partiellen Differentialgleichungen,, in, (1943).

[39]

L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary,, J. Analyse Math., 58 (1992), 94. doi: 10.1007/BF02790229.

[40]

H. Wittich, Ganze Losungen der Differentialgleiehung $\Delta u = e^u$,, Math. Z., 49 (1944), 579. doi: 10.1007/BF01174219.

show all references

References:
[1]

C. Bandle and M. Essen, On the solutions of quasilinear elliptic problems with boundary blow-up,, in, (1994), 93.

[2]

C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations, existence, uniqueness and asymptotic behaviour,, Journal d'analyse math\'ematique, 58 (1992), 9. doi: 10.1007/BF02790355.

[3]

C. Bandle and M. Marcus, Asymptotic behaviour of solutions and their derivatives for semilinear elliptic problems with blowup on the boundary,, Ann. Inst. Henri Poincar\'e, 12 (1995), 155.

[4]

C. Bandle and H. Leutwiler, On a quasilinear elliptic equation and a Riemannian metric invariant under Mobius transformation,, Aequationes Math., 42 (1991), 166. doi: 10.1007/BF01818488.

[5]

C. Bandle and S. Vernier-Piro, Estimates for solutions of quasilinear problems with dead cores,, Za Math. Phys, 54 (2003), 815. doi: 10.1007/s00033-003-3203-4.

[6]

L. Bieberbach, $\Delta u = e^u$ und die automorphen funktionen,, Math. Ann., 77 (1916), 173. doi: 10.1007/BF01456901.

[7]

I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators,, Ann. Fac. Sci Toulouse Math, 13 (2004), 261. doi: 10.5802/afst.1070.

[8]

I. Birindelli and F. Demengel, First eigenvalue and Maximum principle for fully nonlinear singular operators,, Advances in Differential Equations, 11 (2006), 91.

[9]

I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators,, Comm. Pure and Appl. Analysis, 6 (2007), 335. doi: 10.3934/cpaa.2007.6.335.

[10]

I. Birindelli and F. Demengel, The Dirichlet problem for singular fully nonlinear operators,, Discrete and Continuous Dynamical Sytems, (2007), 110.

[11]

I. Birindelli and F. Demengel, Regularity of radial solutions for degenerate fully nonlinear equations,, Arxiv 0339276, (0339).

[12]

I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully non linear operators,, Electronic Journal of Differential Equations, 249 (2010), 1089. doi: 10.1016/j.jde.2010.03.015.

[13]

I. Birindelli and F. Demengel, Uniqueness of the first eigenfunction for fully nonlinear equations: the radial case,, Journal for analysis and its applications, 29 (2010), 75. doi: 10.4171/ZAA/1398.

[14]

I. Birindelli and F. Demengel, Eigenfunctions for singular fully nonlinear equations in unbounded domains,, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 697. doi: 10.1007/s00030-010-0077-y.

[15]

L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations,, Ann. Math., 130 (1989), 189.

[16]

X. Cabre and L. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations F(D2u) = 0,, Topological Meth. Nonlinear Anal., 6 (1995), 31.

[17]

X. Cabre and L. Caffarelli, Interior $C^2$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, , J. Maths Pures Appl., 82 (2003), 573. doi: 10.1016/S0021-7824(03)00029-1.

[18]

I. Capuzzo Dolcetta and A. Vitolo, Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations,, Discrete and Continuous Dynamical Systems (DCDS-A), 28 (2010), 539. doi: 10.3934/dcds.2010.28.539.

[19]

O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry,, J. Differential Equations, 249 (2010), 931. doi: 10.1016/j.jde.2010.02.023.

[20]

O. Costin, L. Dupaigne and O. Goubet, Uniqueness of large solutions,, J. Math. Analysis, 395 (2012), 806. doi: 10.1016/j.jmaa.2012.05.085.

[21]

D. P. Covei, Existence of solutions to quasilinear elliptic problems with boundary blow up,, An. Univ. Oradea Fasc. Mat., 17 (2010).

[22]

G. Davila, P. Felmer and A. Quaas, Harnack Inequality for singular fully nonlinear operators and some existence's results,, Calculus of Variations and PDE, 39 (2010), 557. doi: 10.1007/s00526-010-0325-3.

[23]

M. Del Pino and R. Manasevich, Global bifurcation from the eigenvalues of the p-Laplacian,, Journal of Differential Equations, 92 (1991), 226. doi: 10.1016/0022-0396(91)90048-E.

[24]

G. Diaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness,, Nonlinear Analysis. Theory, 20 (1993), 97. doi: 10.1016/0362-546X(93)90012-H.

[25]

S. Dumont, L. Dupaigne, O. Goubet and V. Radulescu, Back to the Keller-Osserman condition for boundary blow up solutions,, Advanced Nonlinear Studies, 7 (2007), 271.

[26]

M. Esteban, P. Felmer and A. Quaas, Super-linear elliptic equation for fully nonlinear operators without growth restrictions for the data,, Proc. Roy. Soc. Edinburgh, 53 (2010), 125. doi: 10.1017/S0013091507001393.

[27]

C. Imbert, Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate fully non-linear elliptic equations,, Journal of Differential Equations, 250 (2011), 1553. doi: 10.1016/j.jde.2010.07.005.

[28]

J. B. Keller, On solutions of $\Delta u =f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503. doi: 10.1002/cpa.3160100402.

[29]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary value problem,, Proc. Amer. Math. Soc., 111 (1991), 721. doi: 10.1090/S0002-9939-1991-1037213-9.

[30]

C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations,, in, (1974).

[31]

M. Marcus and L. Véron, Uniqueness and asymptotic behaviour of solutions with boundary blowup for a class of nonlinear elliptic equations,, Ann. Inst. Henri Poincar\'e, 14 (1997), 237. doi: 10.1016/S0294-1449(97)80146-1.

[32]

M. Marcus and L. Véron, Existence and uniqueness results for large solutions of nonlinear elliptic equations,, J. Evol. Equations, 3 (2004), 637.

[33]

J. Matero, Quasilinear elliptic equations with boundary blow up,, Journal d'analyse math\'ematique, 69 (1996), 229. doi: 10.1007/BF02787108.

[34]

N. Nadirashvili and S. Vladut, On axially symmetric solutions of fully nonlinear elliptic equations,, To appear in Math. Z., ().

[35]

P. Pucci and J. Serrin, Dead Cores and Bursts for quasilinear Singular elliptic equations,, SIAM Journal of Math. Analysis, 38 (2006), 259. doi: 10.1137/050630027.

[36]

R. Osserman, On the inequality $\Delta u \geq f(u)$,, Pacific J. Math., 7 (1957), 1641.

[37]

S. L. Pohozaev, The Dirichlet problem for the equation $\Delta u = u^2$,, Sov. Math. Dokl., 1 (1961), 1143.

[38]

H. Rademacher, Einige besondere Probleme der partiellen Differentialgleichungen,, in, (1943).

[39]

L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary,, J. Analyse Math., 58 (1992), 94. doi: 10.1007/BF02790229.

[40]

H. Wittich, Ganze Losungen der Differentialgleiehung $\Delta u = e^u$,, Math. Z., 49 (1944), 579. doi: 10.1007/BF01174219.

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