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May  2011, 30(2): 493-508. doi: 10.3934/dcds.2011.30.493

Fundamental solutions for a class of Isaacs integral operators

Patricio Felmer 1, and  Alexander Quaas 2,

1. 

Departamento de Ingeneria Matematica F.C.F.M., Universidad de Chile, Casilla 170 Correro 3, Santiago
2. 

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

Received  May 2010 Published  February 2011

In this article we study the existence of fundamental solutions for a class of Isaacs integral operators and we apply them to prove Liouville type theorems. In proving these theorems we use the comparison principle for non-local operators.
Keywords: fundamental solutions, Nonlinear integral operators, Liouville type theorems..
Mathematics Subject Classification: Primary: 47G20, 45K05; Secondary: 35B50, 35B53, 35J6.
Citation: Patricio Felmer, Alexander Quaas. Fundamental solutions for a class of Isaacs integral operators. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 493-508. doi: 10.3934/dcds.2011.30.493
References:
[1]

S. Armstrong and B. Sirakov, Sharp Liouville results for Fully Nonlinear equations with power-growth nonlinearities, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2010) to appear. Google Scholar

[2]

S. Armstrong, B. Sirakov and C. Smart, Fundamental solutions of fully nonlinear elliptic equations, Communications on Pure and Applied Mathematics (2011) to appear. Google Scholar

[3]

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

[4]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  Google Scholar

[5]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semi-linear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42, 3 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar

[6]

I. Capuzzo-Dolcetta and A. Cutri, Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Communications in Contemporary Mathematics, 3 (2003), 435-448. doi: 10.1142/S0219199703001014.  Google Scholar

[7]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. Journal, 3 (1991), 615-622.  Google Scholar

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDE, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[9]

A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Analyse non lineaire, 17 (2000), 219-245.  Google Scholar

[10]

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

[11]

P. Felmer, A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (2009), 5721-5736. doi: 10.1090/S0002-9947-09-04566-8.  Google Scholar

[12]

P. Felmer and A. Quaas, Fundamental solutions and Liouville type properties for nonlinear integral operators, Advances in Mathematics, 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023.  Google Scholar

[13]

B. Gidas, Symmetry and isolated singularitiesof positive solutions of nonlinear elliptic equations, Nonlinear partial differential equations in engineering and applied science (Proc. Conf., Univ. Rhode Island, Kingston, R.I., 1979), pp. 255-273, Lecture Notes in Pure and Appl. Math., 54, Dekker, New York, 1980.  Google Scholar

[14]

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: 10.1002/cpa.3160340406.  Google Scholar

[15]

D. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214. doi: 10.1007/s002050000108.  Google Scholar

[16]

D. Labutin, Isolated singularities for fully nonlinear elliptic equations, Journal of Differential Equation, 177 (2001), 49-76. doi: 10.1006/jdeq.2001.3998.  Google Scholar

[17]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6.  Google Scholar

show all references

References:
[1]

S. Armstrong and B. Sirakov, Sharp Liouville results for Fully Nonlinear equations with power-growth nonlinearities, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2010) to appear. Google Scholar

[2]

S. Armstrong, B. Sirakov and C. Smart, Fundamental solutions of fully nonlinear elliptic equations, Communications on Pure and Applied Mathematics (2011) to appear. Google Scholar

[3]

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

[4]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  Google Scholar

[5]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semi-linear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42, 3 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar

[6]

I. Capuzzo-Dolcetta and A. Cutri, Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Communications in Contemporary Mathematics, 3 (2003), 435-448. doi: 10.1142/S0219199703001014.  Google Scholar

[7]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. Journal, 3 (1991), 615-622.  Google Scholar

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDE, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[9]

A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Analyse non lineaire, 17 (2000), 219-245.  Google Scholar

[10]

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

[11]

P. Felmer, A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (2009), 5721-5736. doi: 10.1090/S0002-9947-09-04566-8.  Google Scholar

[12]

P. Felmer and A. Quaas, Fundamental solutions and Liouville type properties for nonlinear integral operators, Advances in Mathematics, 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023.  Google Scholar

[13]

B. Gidas, Symmetry and isolated singularitiesof positive solutions of nonlinear elliptic equations, Nonlinear partial differential equations in engineering and applied science (Proc. Conf., Univ. Rhode Island, Kingston, R.I., 1979), pp. 255-273, Lecture Notes in Pure and Appl. Math., 54, Dekker, New York, 1980.  Google Scholar

[14]

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: 10.1002/cpa.3160340406.  Google Scholar

[15]

D. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214. doi: 10.1007/s002050000108.  Google Scholar

[16]

D. Labutin, Isolated singularities for fully nonlinear elliptic equations, Journal of Differential Equation, 177 (2001), 49-76. doi: 10.1006/jdeq.2001.3998.  Google Scholar

[17]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6.  Google Scholar

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