# American Institute of Mathematical Sciences

March  2019, 39(3): 1237-1256. doi: 10.3934/dcds.2019053

## Fundamental solutions of a class of homogeneous integro-differential elliptic equations

 1 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, China 2 Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China 3 Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA

* Corresponding author: NSFC grant 11271236

Received  April 2017 Revised  February 2018 Published  December 2018

Fund Project: The first author is supported by NSFC grant 11126201, 11671243

In this paper, we study a class of integro-differential elliptic operators $L_{σ}$ with kernel $k(y) = a(y)/|y|^{d+σ}$, where $d≥2, σ∈(0,2)$, and the positive function $a(y)$ is homogenous and bounded. By using a purely analytic method, we construct the fundamental solution $Φ$ of $L_{σ}$ if $a(y)$ satisfies a natural cancellation assumption and $|a(y)-1|$ is small. Furthermore, we show that the fundamental solution $Φ$ is $-α^{*}$ homogeneous and Lipschitz continuous, where the constant $α^{*}∈(0,d)$. A Liouville-type theorem demonstrates that the fundamental solution $Φ$ is the unique nontrivial solution of $L_{σ}u = 0$ in $\mathbb{R}^{d}\setminus\{0\}$ that is bounded from below.

Citation: Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053
##### References:
 [1] W. Ao, J. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dynami. Systems, 37 (2017), 5561-5601. doi: 10.3934/dcds.2017242. [2] S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, Comm. Pure Appl. Math., 64 (2011), 737-777. doi: 10.1002/cpa.20360. [3] S. N. Armstrong, B. Sirakov and C. K. Smart, Singular solutions of fully nonlinear elliptic equations and applications, Arch. Rational Mech. Anal., 205 (2013), 345-394. doi: 10.1007/s00205-012-0505-8. [4] R. F. Bass and M. Kassmann, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc., 357 (2005), 837-850. doi: 10.1090/S0002-9947-04-03549-4. [5] M. Bôcher, Singular points of functions which satisfy partial differential equations of the elliptic type, Bull. Amer. Math. Soc., 9 (1903), 455-465. doi: 10.1090/S0002-9904-1903-01017-9. [6] K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. [7] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. Poincaré, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [8] L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213. doi: 10.2307/1971480. [9] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, Vol. 43, Providence, R.I., 1995. doi: 10.1090/coll/043. [10] L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. Ⅰ, J. Fixed point Theory Appl., 5 (2009), 353-395. doi: 10.1007/s11784-009-0107-8. [11] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [12] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274. [13] L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for non local fully non linear equations, Ann. of Math., 174 (2011), 1163-1187. doi: 10.4007/annals.2011.174.2.9. [14] W. Chen, L. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381. doi: 10.1016/j.na.2014.11.003. [15] Y. Chen and C. Wei, Partial regularity of solutions to the fractional Navier-Stokes equations, Discrete Contin. Dynami. Systems, 36 (2016), 5309-5322. doi: 10.3934/dcds.2016033. [16] Z. Chen and X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Relat. Fields, 165 (2016), 267-312. doi: 10.1007/s00440-015-0631-y. [17] T. Cheng, Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dynami. Systems, 37 (2017), 3587-3599. doi: 10.3934/dcds.2017154. [18] H. Dong and D. Kim, On $L^{p}$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199. doi: 10.1016/j.jfa.2011.11.002. [19] H. Dong and D. Kim, Schauder estimates for a class of non-local elliptic equations, Discrete Contin. Dynami. Systems, 33 (2013), 2319-2347. doi: 10.3934/dcds.2013.33.2319. [20] P. Felmer and A. Quaas, Fundamental solutions for a class of Isaacs integral operators, Discrete Contin. Dynami. Systems, 30 (2011), 493-508. doi: 10.3934/dcds.2011.30.493. [21] P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear operators, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023. [22] D. Gilbarg and J. Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math., 4 (1955/56), 309-340. doi: 10.1007/BF02787726. [23] D. Gilbarg and S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0. [24] D. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214. doi: 10.1007/s002050000108. [25] D. Labutin, Isolated singularities for fully nonlinear elliptic equations, J. Differential Equations, 177 (2001), 49-76. doi: 10.1006/jdeq.2001.3998. [26] L. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706. [27] E. M. Stein, Singular Integrals and Differential Property of Functions, Princeton press, Princeton, 1970. [28] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680. [29] L. Véron, Singularities of Solutions of Second Order Quasilinear Equations, Pitman research notes in mathematics Series, 353. Longman. Harlow, 1996. [30] Z. Wang and H. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dynami. Systems, 36 (2016), 499-508. doi: 10.3934/dcds.2016.36.499. [31] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Mathematics, 2 (2014), 423-452. [32] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dynami. Systems, 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.

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##### References:
 [1] W. Ao, J. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dynami. Systems, 37 (2017), 5561-5601. doi: 10.3934/dcds.2017242. [2] S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, Comm. Pure Appl. Math., 64 (2011), 737-777. doi: 10.1002/cpa.20360. [3] S. N. Armstrong, B. Sirakov and C. K. Smart, Singular solutions of fully nonlinear elliptic equations and applications, Arch. Rational Mech. Anal., 205 (2013), 345-394. doi: 10.1007/s00205-012-0505-8. [4] R. F. Bass and M. Kassmann, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc., 357 (2005), 837-850. doi: 10.1090/S0002-9947-04-03549-4. [5] M. Bôcher, Singular points of functions which satisfy partial differential equations of the elliptic type, Bull. Amer. Math. Soc., 9 (1903), 455-465. doi: 10.1090/S0002-9904-1903-01017-9. [6] K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. [7] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. Poincaré, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [8] L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213. doi: 10.2307/1971480. [9] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, Vol. 43, Providence, R.I., 1995. doi: 10.1090/coll/043. [10] L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. Ⅰ, J. Fixed point Theory Appl., 5 (2009), 353-395. doi: 10.1007/s11784-009-0107-8. [11] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [12] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274. [13] L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for non local fully non linear equations, Ann. of Math., 174 (2011), 1163-1187. doi: 10.4007/annals.2011.174.2.9. [14] W. Chen, L. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381. doi: 10.1016/j.na.2014.11.003. [15] Y. Chen and C. Wei, Partial regularity of solutions to the fractional Navier-Stokes equations, Discrete Contin. Dynami. Systems, 36 (2016), 5309-5322. doi: 10.3934/dcds.2016033. [16] Z. Chen and X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Relat. Fields, 165 (2016), 267-312. doi: 10.1007/s00440-015-0631-y. [17] T. Cheng, Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dynami. Systems, 37 (2017), 3587-3599. doi: 10.3934/dcds.2017154. [18] H. Dong and D. Kim, On $L^{p}$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199. doi: 10.1016/j.jfa.2011.11.002. [19] H. Dong and D. Kim, Schauder estimates for a class of non-local elliptic equations, Discrete Contin. Dynami. Systems, 33 (2013), 2319-2347. doi: 10.3934/dcds.2013.33.2319. [20] P. Felmer and A. Quaas, Fundamental solutions for a class of Isaacs integral operators, Discrete Contin. Dynami. Systems, 30 (2011), 493-508. doi: 10.3934/dcds.2011.30.493. [21] P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear operators, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023. [22] D. Gilbarg and J. Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math., 4 (1955/56), 309-340. doi: 10.1007/BF02787726. [23] D. Gilbarg and S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0. [24] D. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214. doi: 10.1007/s002050000108. [25] D. Labutin, Isolated singularities for fully nonlinear elliptic equations, J. Differential Equations, 177 (2001), 49-76. doi: 10.1006/jdeq.2001.3998. [26] L. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706. [27] E. M. Stein, Singular Integrals and Differential Property of Functions, Princeton press, Princeton, 1970. [28] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680. [29] L. Véron, Singularities of Solutions of Second Order Quasilinear Equations, Pitman research notes in mathematics Series, 353. Longman. Harlow, 1996. [30] Z. Wang and H. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dynami. Systems, 36 (2016), 499-508. doi: 10.3934/dcds.2016.36.499. [31] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Mathematics, 2 (2014), 423-452. [32] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dynami. Systems, 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.
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