March 2019, 39(3): 1269-1310. doi: 10.3934/dcds.2019055

Direct methods on fractional equations

1. 

Department of Mathematical Sciences, Yeshiva University, New York NY 10033, USA

2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, China

3. 

Department of Mathematical Sciences, Yeshiva University, USA

Received  January 2018 Revised  May 2018 Published  December 2018

Fund Project: The second author is partially supported Fundamental Research Funds for the Central Universities (lzujbky-2017-it53)

In this paper, we summarize some of the recent developments in the area of fractional equations with focus on the ideas and direct methods on fractional non-local operators. These results have more or less appeared in a series of previous literature, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illustrate the inner connections among them, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and apply them to a variety of problems in this area.

Citation: Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055
References:
[1]

B. BarriosL. Del PezzoJ. Garcia-Melian and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, Revista Matematica Iberoamericana, 34 (2018), 195-220. doi: 10.4171/RMI/983.

[2]

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.

[3]

C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.

[4]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[5]

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.

[6]

M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 4603-4615. doi: 10.3934/dcds.2018201.

[7]

W. ChenL. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Analysis, Theory, Methods & Appl, 121 (2015), 370-381. doi: 10.1016/j.na.2014.11.003.

[8]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198. doi: 10.1016/j.aim.2014.12.013.

[9]

W. Chen and C. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math., 48 (1995), 657-667. doi: 10.1002/cpa.3160480606.

[10]

W. Chen and C. Li, A note on Kazdan-Warner conditions, J. Diff. Geom., 41 (1995), 259-268. doi: 10.4310/jdg/1214456217.

[11]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.

[12]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, vol. 4, 2010.

[13]

W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758. doi: 10.1016/j.aim.2018.07.016.

[14]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Cal. Var. & PDEs, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.

[15]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. in Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038.

[16]

W. Chen, C. Li and Y. Li, A direct blow-up and rescaling argument on nonlocal elliptic equations, International J. Math., 27 (2016), 1650064, 20 pp. doi: 10.1142/S0129167X16500646.

[17]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, a book to be published by World Scientific Publishing Co. 2017.

[18]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, CPAM, 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[19]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347.

[20]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.

[21]

W. ChenY. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157. doi: 10.1016/j.jfa.2017.02.022.

[22]

W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, submitted to the same DCDS issue, 2017.

[23]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Diff. Equa., 260 (2016), 4758-4785. doi: 10.1016/j.jde.2015.11.029.

[24]

T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Disc. Cont. Dyn. Sys. - Series A, 37 (2017), 3587-3599. doi: 10.3934/dcds.2017154.

[25]

J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 3939-3953.

[26]

M. Fall, Entire s-harmonic functions are affine, Proc. AMS, 144 (2016), 2587-2592. doi: 10.1090/proc/13021.

[27]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018.

[28]

R. L. Frank and E. Lieb, Inversion positivityand the sharp Hardy-Littlewood-Sobolev inequality, Cal. Var. & PDEs, 39 (2010), 85-99. doi: 10.1007/s00526-009-0302-x.

[29]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, 2001.

[30]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602.

[31]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.

[32]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.

[33]

S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali Math. Pura et Appl., 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y.

[34]

T. JinY. Y. Li and J. Xiong, On a fractional Nirenberg problem, part Ⅰ: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.

[35]

T. Jin and J. Xiong, A fractional Yemabe flow and some applications, J. reine angew. Math., 696 (2014), 187-223. doi: 10.1515/crelle-2012-0110.

[36]

J. Kazdan, Prescribing the Curvature of a Riemannian Manifold, CBMS Regional Conference Series in Mathematics, 57. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. doi: 10.1090/cbms/057.

[37]

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag Berlin Heidelberg, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.

[38]

Y. Lei, Asymptotic properties of positive solutions of the Hardy Sobolev type equations, J. Diff. Equa., 254 (2013), 1774-1799. doi: 10.1016/j.jde.2012.11.008.

[39]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy Littlewood Sobolev system of integral equations, Cal. Var. & PDEs, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7.

[40]

C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, June 20, 2018.

[41]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke. Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.

[42]

C. S. Lin, On Liouville theorem and apriori estimates for the scalar curvature equations, Ann. Scula Norm. Sup. Pisa CI.Sci., 27 (1998), 107-130.

[43]

B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 5339-5349. doi: 10.3934/dcds.2018235.

[44]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036.

[45]

G. Lu and J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math., 253 (2011), 455-473. doi: 10.2140/pjm.2011.253.455.

[46]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Cal. Var. & PDEs, 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7.

[47]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.

[48]

C. MaW. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Math., 3 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.

[49]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.

[50]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. de Math. Pures et Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.

[51]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.

[52]

X. Xu and P. Yang, Remarks on prescribing Gaussian curvature, Trans. AMS, 336 (1993), 831-840. doi: 10.1090/S0002-9947-1993-1087058-5.

[53]

L. ZhangW. ChenC. Li and T. Cheng, A Liouville theorem for $ \alpha $-harmonic functions in $ \mathbb{R}^n_+ $, Disc. Cont. Dyn. Sys., 36 (2016), 1721-1736. doi: 10.3934/dcds.2016.36.1721.

[54]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.

show all references

References:
[1]

B. BarriosL. Del PezzoJ. Garcia-Melian and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, Revista Matematica Iberoamericana, 34 (2018), 195-220. doi: 10.4171/RMI/983.

[2]

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.

[3]

C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.

[4]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[5]

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.

[6]

M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 4603-4615. doi: 10.3934/dcds.2018201.

[7]

W. ChenL. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Analysis, Theory, Methods & Appl, 121 (2015), 370-381. doi: 10.1016/j.na.2014.11.003.

[8]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198. doi: 10.1016/j.aim.2014.12.013.

[9]

W. Chen and C. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math., 48 (1995), 657-667. doi: 10.1002/cpa.3160480606.

[10]

W. Chen and C. Li, A note on Kazdan-Warner conditions, J. Diff. Geom., 41 (1995), 259-268. doi: 10.4310/jdg/1214456217.

[11]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.

[12]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, vol. 4, 2010.

[13]

W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758. doi: 10.1016/j.aim.2018.07.016.

[14]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Cal. Var. & PDEs, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.

[15]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. in Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038.

[16]

W. Chen, C. Li and Y. Li, A direct blow-up and rescaling argument on nonlocal elliptic equations, International J. Math., 27 (2016), 1650064, 20 pp. doi: 10.1142/S0129167X16500646.

[17]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, a book to be published by World Scientific Publishing Co. 2017.

[18]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, CPAM, 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[19]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347.

[20]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.

[21]

W. ChenY. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157. doi: 10.1016/j.jfa.2017.02.022.

[22]

W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, submitted to the same DCDS issue, 2017.

[23]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Diff. Equa., 260 (2016), 4758-4785. doi: 10.1016/j.jde.2015.11.029.

[24]

T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Disc. Cont. Dyn. Sys. - Series A, 37 (2017), 3587-3599. doi: 10.3934/dcds.2017154.

[25]

J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 3939-3953.

[26]

M. Fall, Entire s-harmonic functions are affine, Proc. AMS, 144 (2016), 2587-2592. doi: 10.1090/proc/13021.

[27]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018.

[28]

R. L. Frank and E. Lieb, Inversion positivityand the sharp Hardy-Littlewood-Sobolev inequality, Cal. Var. & PDEs, 39 (2010), 85-99. doi: 10.1007/s00526-009-0302-x.

[29]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, 2001.

[30]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602.

[31]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.

[32]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.

[33]

S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali Math. Pura et Appl., 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y.

[34]

T. JinY. Y. Li and J. Xiong, On a fractional Nirenberg problem, part Ⅰ: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.

[35]

T. Jin and J. Xiong, A fractional Yemabe flow and some applications, J. reine angew. Math., 696 (2014), 187-223. doi: 10.1515/crelle-2012-0110.

[36]

J. Kazdan, Prescribing the Curvature of a Riemannian Manifold, CBMS Regional Conference Series in Mathematics, 57. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. doi: 10.1090/cbms/057.

[37]

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag Berlin Heidelberg, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.

[38]

Y. Lei, Asymptotic properties of positive solutions of the Hardy Sobolev type equations, J. Diff. Equa., 254 (2013), 1774-1799. doi: 10.1016/j.jde.2012.11.008.

[39]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy Littlewood Sobolev system of integral equations, Cal. Var. & PDEs, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7.

[40]

C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, June 20, 2018.

[41]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke. Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.

[42]

C. S. Lin, On Liouville theorem and apriori estimates for the scalar curvature equations, Ann. Scula Norm. Sup. Pisa CI.Sci., 27 (1998), 107-130.

[43]

B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 5339-5349. doi: 10.3934/dcds.2018235.

[44]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036.

[45]

G. Lu and J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math., 253 (2011), 455-473. doi: 10.2140/pjm.2011.253.455.

[46]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Cal. Var. & PDEs, 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7.

[47]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.

[48]

C. MaW. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Math., 3 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.

[49]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.

[50]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. de Math. Pures et Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.

[51]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.

[52]

X. Xu and P. Yang, Remarks on prescribing Gaussian curvature, Trans. AMS, 336 (1993), 831-840. doi: 10.1090/S0002-9947-1993-1087058-5.

[53]

L. ZhangW. ChenC. Li and T. Cheng, A Liouville theorem for $ \alpha $-harmonic functions in $ \mathbb{R}^n_+ $, Disc. Cont. Dyn. Sys., 36 (2016), 1721-1736. doi: 10.3934/dcds.2016.36.1721.

[54]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.

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