January  2011, 29(1): 51-65. doi: 10.3934/dcds.2011.29.51

Multiplicity results for extremal operators through bifurcation

1. 

Departamento de Matemática, Universidad Técnico Fedrico Santa María, Avenida España 1680, Casilla 110-V, Valparaíso, Chile, Chile

Received  August 2009 Revised  June 2010 Published  September 2010

We study non-proper uniformly elliptic fully nonlinear equations involving extremal operators of Pucci type. We prove the existence of all radial spectrum for this type of operators and establish a multiplicity existence results through global bifurcation.
Citation: Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51
References:
[1]

M. Arias and J. Campos, Radial Fucik spectrum of the Laplace operator, Journal of Mathematical Analysis and Applications, 190 (1995), 654-666. doi: doi:10.1006/jmaa.1995.1101.  Google Scholar

[2]

I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators, Commun. Pure Appl. Anal., 6 (2007), 335-366. doi: doi:10.3934/cpaa.2007.6.335.  Google Scholar

[3]

I. Birindelli and F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differential Equations, 11 (2006), 91-119.  Google Scholar

[4]

J. Busca, M. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operator, Ann. Inst. Henri Poincare (C) Non Linear Analysis, 22 (2005), 187-206.  Google Scholar

[5]

X. Cabré and L. Caffarelli, "Fully Nonlinear Elliptic Equation," American Mathematical Society, Colloquium Publication, 43, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[6]

L. A. Caffarelli, M. G. Crandall, M. Kocan and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397. doi: doi:10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A.  Google Scholar

[7]

L. Caffarelli, J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations II, Comm. Pure Appl. Math., 38 (1985), 209-252. doi: doi:10.1002/cpa.3160380206.  Google Scholar

[8]

M. G. Crandall, M. Kocan, P. L. Lions and A.Świech, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Elec. J. Diff. Eq., 24 (1999), 1-20.  Google Scholar

[9]

M. G. Crandall, M. Kocan and A. Świech, $L^p$ theory for fully nonlinear uniformly parabolic equations, Comm. Part. Diff. Eq., 25 (2000), 1997-2053.  Google Scholar

[10]

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. doi: doi:10.1090/S0273-0979-1992-00266-5.  Google Scholar

[11]

F. Da Lio and B. Sirakov, Symmetry results for viscosity solution of fully nonlinear elliptic equation, J. Eur. Math. Soc., 9 (2006), 317-330.  Google Scholar

[12]

P. Drábek, "Solvability and Bifurcations of Nonlinear Equations," Pitman Research Notes in Mathematics Series, 264, Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc., New York, 1992.  Google Scholar

[13]

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 25 (1982), 333-363. doi: doi:10.1002/cpa.3160350303.  Google Scholar

[14]

P. Felmer and A. Quaas, Positive solutions to 'semi-linear' equation involving the Pucci's operator, J. Diff. Eqs., 199 (2004), 376-393.  Google Scholar

[15]

P. Felmer and A. Quaas, Critical exponents for uniformly elliptic extremal operators, Indiana Univ. Math. J., 55 (2006), 593-629. doi: doi:10.1512/iumj.2006.55.2864.  Google Scholar

[16]

P. Felmer and A. Quaas, On fundamental solutions, dimension and two properties of uniformly elliptic maximal operators, Trans. of the American Math. Society, 361 (2009), 5721-5736. doi: doi:10.1090/S0002-9947-09-04566-8.  Google Scholar

[17]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: doi:10.1002/cpa.3160340406.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equation of Second Order," 2nd edition Springer-verlag, 1983.  Google Scholar

[19]

P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Diff. Eqs., 236 (2007), 532-550.  Google Scholar

[20]

P. L. Lions, Resolution analytique des problemes de Bellman-Dirichlet, Acta Math., 146 (1981), 151-166. doi: doi:10.1007/BF02392461.  Google Scholar

[21]

P. L. Lions, Bifurcation and optimal stochastic control, Nonlinear Anal., 2 (1983), 177-207. doi: doi:10.1016/0362-546X(83)90081-0.  Google Scholar

[22]

C. Pucci, Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Amer. Math. Soc., 17 (1966), 788-795.  Google Scholar

[23]

C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pure Appl., 72 (1966), 141-170. doi: doi:10.1007/BF02414332.  Google Scholar

[24]

A. Quaas, Existence of positive solutions to a 'semilinear' equation involving the Pucci's operator in a convex domain, Diff. Integral Equation, 17 (2004), 481-494.  Google Scholar

[25]

A. Quaas and B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci operator, Comm. Part. Diff. Eq., 31 (2006), 987-1003.  Google Scholar

[26]

A. Quaas and B. Sirakov, On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Comptes Rendus Mathematique, 342 (2006), 115-118. doi: doi:10.1016/j.crma.2005.11.003.  Google Scholar

[27]

A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., 218 (2008), 105-135. doi: doi:10.1016/j.aim.2007.12.002.  Google Scholar

[28]

P. H. Rabinowitz, Some aspect of nonlinear eigenvalue problem, Rocky Moutain J. Math., 74 (1973), 161-202. doi: doi:10.1216/RMJ-1973-3-2-161.  Google Scholar

[29]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: doi:10.1016/0022-1236(71)90030-9.  Google Scholar

[30]

P. H. Rabinowitz, "Théorie du Degré Topologique et Applications à des Problèmes aux Limites non Linéaires," Lectures Notes Lab. Analyse Numérique Université PARIS VI, 1975. Google Scholar

[31]

A. Świech, $W^{1,p}$-estimates for solutions of fully nonlinear uniformly elliptic equations, Adv. Diff. Eq., 2 (1997), 1005-1027.  Google Scholar

show all references

References:
[1]

M. Arias and J. Campos, Radial Fucik spectrum of the Laplace operator, Journal of Mathematical Analysis and Applications, 190 (1995), 654-666. doi: doi:10.1006/jmaa.1995.1101.  Google Scholar

[2]

I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators, Commun. Pure Appl. Anal., 6 (2007), 335-366. doi: doi:10.3934/cpaa.2007.6.335.  Google Scholar

[3]

I. Birindelli and F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differential Equations, 11 (2006), 91-119.  Google Scholar

[4]

J. Busca, M. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operator, Ann. Inst. Henri Poincare (C) Non Linear Analysis, 22 (2005), 187-206.  Google Scholar

[5]

X. Cabré and L. Caffarelli, "Fully Nonlinear Elliptic Equation," American Mathematical Society, Colloquium Publication, 43, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[6]

L. A. Caffarelli, M. G. Crandall, M. Kocan and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397. doi: doi:10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A.  Google Scholar

[7]

L. Caffarelli, J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations II, Comm. Pure Appl. Math., 38 (1985), 209-252. doi: doi:10.1002/cpa.3160380206.  Google Scholar

[8]

M. G. Crandall, M. Kocan, P. L. Lions and A.Świech, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Elec. J. Diff. Eq., 24 (1999), 1-20.  Google Scholar

[9]

M. G. Crandall, M. Kocan and A. Świech, $L^p$ theory for fully nonlinear uniformly parabolic equations, Comm. Part. Diff. Eq., 25 (2000), 1997-2053.  Google Scholar

[10]

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. doi: doi:10.1090/S0273-0979-1992-00266-5.  Google Scholar

[11]

F. Da Lio and B. Sirakov, Symmetry results for viscosity solution of fully nonlinear elliptic equation, J. Eur. Math. Soc., 9 (2006), 317-330.  Google Scholar

[12]

P. Drábek, "Solvability and Bifurcations of Nonlinear Equations," Pitman Research Notes in Mathematics Series, 264, Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc., New York, 1992.  Google Scholar

[13]

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 25 (1982), 333-363. doi: doi:10.1002/cpa.3160350303.  Google Scholar

[14]

P. Felmer and A. Quaas, Positive solutions to 'semi-linear' equation involving the Pucci's operator, J. Diff. Eqs., 199 (2004), 376-393.  Google Scholar

[15]

P. Felmer and A. Quaas, Critical exponents for uniformly elliptic extremal operators, Indiana Univ. Math. J., 55 (2006), 593-629. doi: doi:10.1512/iumj.2006.55.2864.  Google Scholar

[16]

P. Felmer and A. Quaas, On fundamental solutions, dimension and two properties of uniformly elliptic maximal operators, Trans. of the American Math. Society, 361 (2009), 5721-5736. doi: doi:10.1090/S0002-9947-09-04566-8.  Google Scholar

[17]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: doi:10.1002/cpa.3160340406.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equation of Second Order," 2nd edition Springer-verlag, 1983.  Google Scholar

[19]

P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Diff. Eqs., 236 (2007), 532-550.  Google Scholar

[20]

P. L. Lions, Resolution analytique des problemes de Bellman-Dirichlet, Acta Math., 146 (1981), 151-166. doi: doi:10.1007/BF02392461.  Google Scholar

[21]

P. L. Lions, Bifurcation and optimal stochastic control, Nonlinear Anal., 2 (1983), 177-207. doi: doi:10.1016/0362-546X(83)90081-0.  Google Scholar

[22]

C. Pucci, Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Amer. Math. Soc., 17 (1966), 788-795.  Google Scholar

[23]

C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pure Appl., 72 (1966), 141-170. doi: doi:10.1007/BF02414332.  Google Scholar

[24]

A. Quaas, Existence of positive solutions to a 'semilinear' equation involving the Pucci's operator in a convex domain, Diff. Integral Equation, 17 (2004), 481-494.  Google Scholar

[25]

A. Quaas and B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci operator, Comm. Part. Diff. Eq., 31 (2006), 987-1003.  Google Scholar

[26]

A. Quaas and B. Sirakov, On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Comptes Rendus Mathematique, 342 (2006), 115-118. doi: doi:10.1016/j.crma.2005.11.003.  Google Scholar

[27]

A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., 218 (2008), 105-135. doi: doi:10.1016/j.aim.2007.12.002.  Google Scholar

[28]

P. H. Rabinowitz, Some aspect of nonlinear eigenvalue problem, Rocky Moutain J. Math., 74 (1973), 161-202. doi: doi:10.1216/RMJ-1973-3-2-161.  Google Scholar

[29]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: doi:10.1016/0022-1236(71)90030-9.  Google Scholar

[30]

P. H. Rabinowitz, "Théorie du Degré Topologique et Applications à des Problèmes aux Limites non Linéaires," Lectures Notes Lab. Analyse Numérique Université PARIS VI, 1975. Google Scholar

[31]

A. Świech, $W^{1,p}$-estimates for solutions of fully nonlinear uniformly elliptic equations, Adv. Diff. Eq., 2 (1997), 1005-1027.  Google Scholar

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