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:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T. Cheng, Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dynami. Systems, 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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D. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214.  doi: 10.1007/s002050000108.  Google Scholar

[25]

D. Labutin, Isolated singularities for fully nonlinear elliptic equations, J. Differential Equations, 177 (2001), 49-76.  doi: 10.1006/jdeq.2001.3998.  Google Scholar

[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.  Google Scholar

[27]

E. M. Stein, Singular Integrals and Differential Property of Functions, Princeton press, Princeton, 1970.  Google Scholar

[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.  Google Scholar

[29]

L. Véron, Singularities of Solutions of Second Order Quasilinear Equations, Pitman research notes in mathematics Series, 353. Longman. Harlow, 1996.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. Labutin, Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214.  doi: 10.1007/s002050000108.  Google Scholar

[25]

D. Labutin, Isolated singularities for fully nonlinear elliptic equations, J. Differential Equations, 177 (2001), 49-76.  doi: 10.1006/jdeq.2001.3998.  Google Scholar

[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.  Google Scholar

[27]

E. M. Stein, Singular Integrals and Differential Property of Functions, Princeton press, Princeton, 1970.  Google Scholar

[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.  Google Scholar

[29]

L. Véron, Singularities of Solutions of Second Order Quasilinear Equations, Pitman research notes in mathematics Series, 353. Longman. Harlow, 1996.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

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