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. AoJ. 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. ArmstrongB. 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. ArmstrongB. 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.

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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.

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K. BogdanT. 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.

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L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213. doi: 10.2307/1971480.

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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. CaffarelliY. 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. ChenL. 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. ZhuoW. ChenX. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Mathematics, 2 (2014), 423-452.

[32]

R. ZhuoW. ChenX. 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.

show all references

References:
[1]

W. AoJ. 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. ArmstrongB. 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. ArmstrongB. 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. BogdanT. 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. CaffarelliY. 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. ChenL. 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. ZhuoW. ChenX. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Mathematics, 2 (2014), 423-452.

[32]

R. ZhuoW. ChenX. 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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