Fundamental solutions for a class of Isaacs integral operators

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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

Abstract
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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 and 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.
[2]	S. Armstrong, B. Sirakov and C. Smart, Fundamental solutions of fully nonlinear elliptic equations, Communications on Pure and Applied Mathematics (2011) to appear.
[3]	X. Cabré and L. Caffarelli, "Fully Nonlinear Elliptic Equation," American Mathematical Society, Colloquium Publication, Vol. 43, 1995.
[4]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
[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.
[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.
[7]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. Journal, 3 (1991), 615-622.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	D. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214. doi: 10.1007/s002050000108.
[16]	D. Labutin, Isolated singularities for fully nonlinear elliptic equations, Journal of Differential Equation, 177 (2001), 49-76. doi: 10.1006/jdeq.2001.3998.
[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.

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.
[2]	S. Armstrong, B. Sirakov and C. Smart, Fundamental solutions of fully nonlinear elliptic equations, Communications on Pure and Applied Mathematics (2011) to appear.
[3]	X. Cabré and L. Caffarelli, "Fully Nonlinear Elliptic Equation," American Mathematical Society, Colloquium Publication, Vol. 43, 1995.
[4]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
[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.
[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.
[7]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. Journal, 3 (1991), 615-622.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	D. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214. doi: 10.1007/s002050000108.
[16]	D. Labutin, Isolated singularities for fully nonlinear elliptic equations, Journal of Differential Equation, 177 (2001), 49-76. doi: 10.1006/jdeq.2001.3998.
[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.

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