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On finite energy solutions of fractional order equations of the Choquard type. (English) Zbl 1408.35043

Summary: Finite energy solutions are the important class of solutions of the Choquard equation. This paper is concerned with the regularity of weak finite energy solutions. For nonlocal fractional-order equations, an integral system involving the Riesz potential and the Bessel potential plays a key role. Applying the regularity lifting lemma to this integral system, we can see that some weak integrable solution has the better regularity properties. In addition, we also show the relation between such an integrable solution and the finite energy solution. Based on these results, we prove that the weak finite energy solution is also the classical solution under some conditions. Finally, we point out that the least energy with the critical exponent can be represented by the sharp constant of some inequality of Sobolev type though the ground state solution cannot be found.

MSC:

35J60 Nonlinear elliptic equations
35R11 Fractional partial differential equations
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[1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037
[2] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
[3] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010. · Zbl 1214.35023
[4] W. Chen; C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24, 1167-1184, (2009) · Zbl 1176.35067
[5] W. Chen; C. Li; Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308, 404-437, (2017) · Zbl 1362.35320
[6] W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18 pp.
[7] W. Chen; C. Li; B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59, 330-343, (2006) · Zbl 1093.45001
[8] Y. Chen; H. Gao, The Cauchy problem for the Hartree equations under random influences, J. Differential Equations, 259, 5192-5219, (2015) · Zbl 1326.60090
[9] S. Cingolani; M. Clapp; S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. S., 6, 891-908, (2013) · Zbl 1260.35198
[10] S. Cingolani; M. Clapp; S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63, 233-248, (2012) · Zbl 1247.35141
[11] S. Cingolani; S. Secchi; M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140, 973-1009, (2010) · Zbl 1215.35146
[12] W. Dai; J. Huang; Y. Qin; B. Wang; Y. Fang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39, 1389-1403, (2019)
[13] P. Felmer; A. Quaas; J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142, 1237-1262, (2012) · Zbl 1290.35308
[14] X. Han; G. Lu, Regularity of solutions to an integral equation associated with Bessel potential, Commun. Pure Appl. Anal., 10, 1111-1119, (2011) · Zbl 1237.45002
[15] S. Ibrahim; N. Masmoudi; K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4, 405-460, (2011) · Zbl 1270.35132
[16] C. Jin; C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26, 447-457, (2006) · Zbl 1113.45006
[17] Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273, 883-905, (2013) · Zbl 1267.45010
[18] Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45, 388-406, (2013) · Zbl 1277.45007
[19] Y. Lei; C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36, 3277-3315, (2016) · Zbl 1336.35092
[20] Y. Lei; C. Li; C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45, 43-61, (2012) · Zbl 1257.45005
[21] C. Li; L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40, 1049-1057, (2008) · Zbl 1167.35347
[22] C. Li; Z. Wu; H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, 115, 6976-6979, (2018)
[23] Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6, 153-180, (2004) · Zbl 1075.45006
[24] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math., 57, 93-105, (1976/77) · Zbl 0369.35022
[25] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118, 349-374, (1983) · Zbl 0527.42011
[26] E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Rhode Island, 2001. · Zbl 0966.26002
[27] E. Lieb; B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53, 185-194, (1977)
[28] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072, (1980) · Zbl 0453.47042
[29] C. Ma; W. Chen; C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226, 2676-2699, (2011) · Zbl 1209.45006
[30] L. Ma; D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342, 943-949, (2008) · Zbl 1140.45004
[31] L. Ma; L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195, 455-467, (2010) · Zbl 1185.35260
[32] C. Miao; G. Xu; L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl., 91, 49-79, (2009) · Zbl 1154.35078
[33] V. Moroz; J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differential Equations, 254, 3089-3145, (2013) · Zbl 1266.35083
[34] V. Moroz; J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 153-184, (2013) · Zbl 1285.35048
[35] X. Shang; J. Zhang, Multi-peak positive solutions for a fractional nonlinear elliptic equation, Discrete Contin. Dyn. Syst., 35, 3183-3201, (2015) · Zbl 1383.35247
[36] Z. Shen, F. Gao and M. Yang, Multiple solutions for nonhomogeneous Choquard equation involving Hardy-Littlewood-Sobolev critical exponent Z. Angew. Math. Phys. , 68 (2017), Art. 61, 25 pp. · Zbl 1375.35146
[37] E. Stein, Singular Integrals and Differentiability Properties of Function, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. · Zbl 0207.13501
[38] S. Sun; Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263, 3857-3882, (2012) · Zbl 1260.45004
[39] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87, 567-576, (1983) · Zbl 0527.35023
[40] W. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math. Vol. 120, Springer-Verlag, New York, 1989. · Zbl 0692.46022
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