March  2019, 39(3): 1345-1358. doi: 10.3934/dcds.2019057

Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians

Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA

* Corresponding author: Changfeng Gui

Received  February 2018 Revised  June 2018 Published  December 2018

Fund Project: The authors are supported by NSF DMS-1601885

It is known that the supercritical Hardy-Littlewood-Sobolev (HLS) systems with an integer power of Laplacian admit classic solutions. In this paper, we prove that the supercritical HLS systems with fractional Laplacians $ (-Δ)^s $, $ s∈(0,1) $, also admit classic solutions.

Citation: Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057
References:
[1]

A. Abrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies In Advanced Mathematics, 2007. doi: 10.1017/CBO9780511618260. Google Scholar

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[4]

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. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Communications in Contemporary Mathematics, 19 (2017), 1750018, 12pp. doi: 10.1142/S0219199717500183. Google Scholar

[9]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system, arXiv preprint, arXiv: 1412.7275, 2014.Google Scholar

[10]

Z. ChengG. Huang and C. Li, On the Hardy-Littlewood-Sobolev type systems, Commuications on Pure and Applied Analysis, 15 (2016), 2059-2074. doi: 10.3934/cpaa.2016027. Google Scholar

[11]

M. ChipotM. ChlebíkM. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $ \mathbb{R}_+^n $ with a nonlinear boundary condition, Journal of Mathematical Analysis and Applications, 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958. Google Scholar

[12]

W. Y. Ding and W. M. Ni, On the elliptic equation $ {Δ} u+ {K}u^{(n+2)/(n-2)} = 0 $ and related topics, Duke Math J., 52 (1985), 485-506. doi: 10.1215/S0012-7094-85-05224-X. Google Scholar

[13]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125. Google Scholar

[14]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885. doi: 10.4310/MRL.2016.v23.n3.a14. Google Scholar

[15]

C. GuiW. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $ \mathbb{R}^n $, Communications on Pure and Applied Mathematics, 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906. Google Scholar

[16]

C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems, Proceedings of the American Mathematical Society, 144 (2016), 3731-3740. doi: 10.1090/proc/13166. Google Scholar

[17]

C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth, Communications in Partial Differential Equations, 41 (2016), 1029-1039. doi: 10.1080/03605302.2016.1190376. Google Scholar

[18]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. Google Scholar

[19]

J. LiuY. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, Journal of Partial Differential Equations, 19 (2006), 256-270. Google Scholar

[20]

Y. Lü and C. Zhou, Symmetry for an integral system with general nonlinearity, Discrete & Continuous Dynamical Systems-A, 38 (2018), 2867-2877. Google Scholar

[21]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $ R^N $, Differ. Integral Equations, 9 (1996), 465-479. Google Scholar

[22]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[23]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Notes by R. A. Artino The American Mathematical Society, Vol. 6 of Courant Lecture Notes in Mathematics, 2001. doi: 10.1090/cln/006. Google Scholar

[24]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Springer, 2007. Google Scholar

[25]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. Google Scholar

[26]

M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math, 1938, 1-42.Google Scholar

[27]

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

[28]

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628, arXiv: 1207.5986. doi: 10.1007/s00205-014-0740-2. Google Scholar

[29]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653. Google Scholar

[30]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semi. Mat. Fis. Univ. Modena, 46 (1998), 369-380. Google Scholar

[31]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal, 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445. Google Scholar

[32]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105. Google Scholar

[33]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. Google Scholar

[34]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. Google Scholar

[35]

E. Valdinoci, From the long jump random walk to the fractional Laplacian. arXiv preprint, arXiv: 0901.3261, 2009.Google Scholar

[36]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional laplacian, Discrete & Continuous Dynamical Systems-A, 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125. Google Scholar

show all references

References:
[1]

A. Abrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies In Advanced Mathematics, 2007. doi: 10.1017/CBO9780511618260. Google Scholar

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[4]

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. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Communications in Contemporary Mathematics, 19 (2017), 1750018, 12pp. doi: 10.1142/S0219199717500183. Google Scholar

[9]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system, arXiv preprint, arXiv: 1412.7275, 2014.Google Scholar

[10]

Z. ChengG. Huang and C. Li, On the Hardy-Littlewood-Sobolev type systems, Commuications on Pure and Applied Analysis, 15 (2016), 2059-2074. doi: 10.3934/cpaa.2016027. Google Scholar

[11]

M. ChipotM. ChlebíkM. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $ \mathbb{R}_+^n $ with a nonlinear boundary condition, Journal of Mathematical Analysis and Applications, 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958. Google Scholar

[12]

W. Y. Ding and W. M. Ni, On the elliptic equation $ {Δ} u+ {K}u^{(n+2)/(n-2)} = 0 $ and related topics, Duke Math J., 52 (1985), 485-506. doi: 10.1215/S0012-7094-85-05224-X. Google Scholar

[13]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125. Google Scholar

[14]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885. doi: 10.4310/MRL.2016.v23.n3.a14. Google Scholar

[15]

C. GuiW. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $ \mathbb{R}^n $, Communications on Pure and Applied Mathematics, 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906. Google Scholar

[16]

C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems, Proceedings of the American Mathematical Society, 144 (2016), 3731-3740. doi: 10.1090/proc/13166. Google Scholar

[17]

C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth, Communications in Partial Differential Equations, 41 (2016), 1029-1039. doi: 10.1080/03605302.2016.1190376. Google Scholar

[18]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. Google Scholar

[19]

J. LiuY. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, Journal of Partial Differential Equations, 19 (2006), 256-270. Google Scholar

[20]

Y. Lü and C. Zhou, Symmetry for an integral system with general nonlinearity, Discrete & Continuous Dynamical Systems-A, 38 (2018), 2867-2877. Google Scholar

[21]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $ R^N $, Differ. Integral Equations, 9 (1996), 465-479. Google Scholar

[22]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[23]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Notes by R. A. Artino The American Mathematical Society, Vol. 6 of Courant Lecture Notes in Mathematics, 2001. doi: 10.1090/cln/006. Google Scholar

[24]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Springer, 2007. Google Scholar

[25]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. Google Scholar

[26]

M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math, 1938, 1-42.Google Scholar

[27]

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

[28]

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628, arXiv: 1207.5986. doi: 10.1007/s00205-014-0740-2. Google Scholar

[29]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653. Google Scholar

[30]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semi. Mat. Fis. Univ. Modena, 46 (1998), 369-380. Google Scholar

[31]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal, 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445. Google Scholar

[32]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105. Google Scholar

[33]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. Google Scholar

[34]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. Google Scholar

[35]

E. Valdinoci, From the long jump random walk to the fractional Laplacian. arXiv preprint, arXiv: 0901.3261, 2009.Google Scholar

[36]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional laplacian, Discrete & Continuous Dynamical Systems-A, 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125. Google Scholar

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