American Institute of Mathematical Sciences

Advanced
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 - 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

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, (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

[1]

Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017
[2]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565
[3]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[4]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
[5]

Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399
[6]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665
[7]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248
[8]

Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206
[9]

Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719
[10]

Linfen Cao, Wenxiong Chen. Liouville type theorems for poly-harmonic Navier problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3937-3955. doi: 10.3934/dcds.2013.33.3937
[11]

Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401
[12]

M. Á. Burgos-Pérez, J. García-Melián, A. Quaas. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4703-4721. doi: 10.3934/dcds.2016004
[13]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511
[14]

Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011
[15]

Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025
[16]

Yutian Lei, Congming Li. Sharp criteria of Liouville type for some nonlinear systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3277-3315. doi: 10.3934/dcds.2016.36.3277
[17]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006
[18]

Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155
[19]

Dongho Chae, Shangkun Weng. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5267-5285. doi: 10.3934/dcds.2016031
[20]

Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Networks & Heterogeneous Media, 2012, 7 (4) : 967-988. doi: 10.3934/nhm.2012.7.967

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences