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 & Continuous Dynamical Systems - A, 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, (2010). 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., (2011). Google Scholar
[3]	X. Cabré and L. Caffarelli, "Fully Nonlinear Elliptic Equation,", American Mathematical Society, 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. 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. 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. 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. Google Scholar
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. PDE, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar
[9]	A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations,, Ann. Inst. H. Poincar\'e Analyse non lineaire, 17 (2000), 219. Google Scholar
[10]	P. Felmer and A. Quaas, Critical exponents for uniformly elliptic extremal operators,, Indiana Univ. Math. J., 55 (2006), 593. 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. 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. 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., (1979), 255. 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. doi: 10.1002/cpa.3160340406. Google Scholar
[15]	D. Labutin, Removable singularities for fully nonlinear elliptic equations,, Arch. Rational Mech. Anal., 155 (2000), 201. doi: 10.1007/s002050000108. Google Scholar
[16]	D. Labutin, Isolated singularities for fully nonlinear elliptic equations,, Journal of Differential Equation, 177 (2001), 49. 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. doi: 10.4171/JEMS/6. Google Scholar

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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, (2010). 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., (2011). Google Scholar
[3]	X. Cabré and L. Caffarelli, "Fully Nonlinear Elliptic Equation,", American Mathematical Society, 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. 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. 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. 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. Google Scholar
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. PDE, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar
[9]	A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations,, Ann. Inst. H. Poincar\'e Analyse non lineaire, 17 (2000), 219. Google Scholar
[10]	P. Felmer and A. Quaas, Critical exponents for uniformly elliptic extremal operators,, Indiana Univ. Math. J., 55 (2006), 593. 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. 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. 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., (1979), 255. 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. doi: 10.1002/cpa.3160340406. Google Scholar
[15]	D. Labutin, Removable singularities for fully nonlinear elliptic equations,, Arch. Rational Mech. Anal., 155 (2000), 201. doi: 10.1007/s002050000108. Google Scholar
[16]	D. Labutin, Isolated singularities for fully nonlinear elliptic equations,, Journal of Differential Equation, 177 (2001), 49. 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. doi: 10.4171/JEMS/6. Google Scholar

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