# American Institute of Mathematical Sciences

March  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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] L. Bieberbach, $\Delta u = e^u$ und die automorphen funktionen,, Math. Ann., 77 (1916), 173. doi: 10.1007/BF01456901. Google Scholar [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. Google Scholar [8] I. Birindelli and F. Demengel, First eigenvalue and Maximum principle for fully nonlinear singular operators,, Advances in Differential Equations, 11 (2006), 91. Google Scholar [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. Google Scholar [10] I. Birindelli and F. Demengel, The Dirichlet problem for singular fully nonlinear operators,, Discrete and Continuous Dynamical Sytems, (2007), 110. Google Scholar [11] I. Birindelli and F. Demengel, Regularity of radial solutions for degenerate fully nonlinear equations,, Arxiv 0339276, (0339). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations,, Ann. Math., 130 (1989), 189. Google Scholar [16] X. Cabre and L. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations F(D2u) = 0,, Topological Meth. Nonlinear Anal., 6 (1995), 31. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] D. P. Covei, Existence of solutions to quasilinear elliptic problems with boundary blow up,, An. Univ. Oradea Fasc. Mat., 17 (2010). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] J. B. Keller, On solutions of $\Delta u =f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503. doi: 10.1002/cpa.3160100402. Google Scholar [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. Google Scholar [30] C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations,, in, (1974). Google Scholar [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. Google Scholar [32] M. Marcus and L. Véron, Existence and uniqueness results for large solutions of nonlinear elliptic equations,, J. Evol. Equations, 3 (2004), 637. Google Scholar [33] J. Matero, Quasilinear elliptic equations with boundary blow up,, Journal d'analyse math\'ematique, 69 (1996), 229. doi: 10.1007/BF02787108. Google Scholar [34] N. Nadirashvili and S. Vladut, On axially symmetric solutions of fully nonlinear elliptic equations,, To appear in Math. Z., (). Google Scholar [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. Google Scholar [36] R. Osserman, On the inequality $\Delta u \geq f(u)$,, Pacific J. Math., 7 (1957), 1641. Google Scholar [37] S. L. Pohozaev, The Dirichlet problem for the equation $\Delta u = u^2$,, Sov. Math. Dokl., 1 (1961), 1143. Google Scholar [38] H. Rademacher, Einige besondere Probleme der partiellen Differentialgleichungen,, in, (1943). Google Scholar [39] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary,, J. Analyse Math., 58 (1992), 94. doi: 10.1007/BF02790229. Google Scholar [40] H. Wittich, Ganze Losungen der Differentialgleiehung $\Delta u = e^u$,, Math. Z., 49 (1944), 579. doi: 10.1007/BF01174219. Google Scholar

show all references

##### References:
 [1] C. Bandle and M. Essen, On the solutions of quasilinear elliptic problems with boundary blow-up,, in, (1994), 93. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] L. Bieberbach, $\Delta u = e^u$ und die automorphen funktionen,, Math. Ann., 77 (1916), 173. doi: 10.1007/BF01456901. Google Scholar [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. Google Scholar [8] I. Birindelli and F. Demengel, First eigenvalue and Maximum principle for fully nonlinear singular operators,, Advances in Differential Equations, 11 (2006), 91. Google Scholar [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. Google Scholar [10] I. Birindelli and F. Demengel, The Dirichlet problem for singular fully nonlinear operators,, Discrete and Continuous Dynamical Sytems, (2007), 110. Google Scholar [11] I. Birindelli and F. Demengel, Regularity of radial solutions for degenerate fully nonlinear equations,, Arxiv 0339276, (0339). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations,, Ann. Math., 130 (1989), 189. Google Scholar [16] X. Cabre and L. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations F(D2u) = 0,, Topological Meth. Nonlinear Anal., 6 (1995), 31. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] D. P. Covei, Existence of solutions to quasilinear elliptic problems with boundary blow up,, An. Univ. Oradea Fasc. Mat., 17 (2010). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] J. B. Keller, On solutions of $\Delta u =f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503. doi: 10.1002/cpa.3160100402. Google Scholar [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. Google Scholar [30] C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations,, in, (1974). Google Scholar [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. Google Scholar [32] M. Marcus and L. Véron, Existence and uniqueness results for large solutions of nonlinear elliptic equations,, J. Evol. Equations, 3 (2004), 637. Google Scholar [33] J. Matero, Quasilinear elliptic equations with boundary blow up,, Journal d'analyse math\'ematique, 69 (1996), 229. doi: 10.1007/BF02787108. Google Scholar [34] N. Nadirashvili and S. Vladut, On axially symmetric solutions of fully nonlinear elliptic equations,, To appear in Math. Z., (). Google Scholar [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. Google Scholar [36] R. Osserman, On the inequality $\Delta u \geq f(u)$,, Pacific J. Math., 7 (1957), 1641. Google Scholar [37] S. L. Pohozaev, The Dirichlet problem for the equation $\Delta u = u^2$,, Sov. Math. Dokl., 1 (1961), 1143. Google Scholar [38] H. Rademacher, Einige besondere Probleme der partiellen Differentialgleichungen,, in, (1943). Google Scholar [39] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary,, J. Analyse Math., 58 (1992), 94. doi: 10.1007/BF02790229. Google Scholar [40] H. Wittich, Ganze Losungen der Differentialgleiehung $\Delta u = e^u$,, Math. Z., 49 (1944), 579. doi: 10.1007/BF01174219. Google Scholar
 [1] 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 [2] Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671 [3] 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 [4] 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 [5] 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 [6] 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 [7] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [8] Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267 [9] 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 [10] 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 [11] 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 [12] Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583 [13] 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 [14] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [15] Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271 [16] Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683 [17] Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669 [18] Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677 [19] 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 [20] Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126

2018 Impact Factor: 0.925