# American Institute of Mathematical Sciences

March  2019, 39(3): 1497-1515. doi: 10.3934/dcds.2019064

## On finite energy solutions of fractional order equations of the Choquard type

 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Received  December 2017 Revised  April 2018 Published  December 2018

Fund Project: The research was supported by NSF of China (No. 11471164, 11871278, 11671209).

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.

Citation: Yutian Lei. On finite energy solutions of fractional order equations of the Choquard type. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1497-1515. doi: 10.3934/dcds.2019064
##### References:
 [1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443.  doi: 10.1007/s00209-004-0663-y. [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. doi: 10.1090/cln/010. [3] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010. [4] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167. [5] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038. [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. doi: 10.1007/s00526-017-1110-3. [7] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [8] Y. Chen and H. Gao, The Cauchy problem for the Hartree equations under random influences, J. Differential Equations, 259 (2015), 5192-5219.  doi: 10.1016/j.jde.2015.06.021. [9] S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. S., 6 (2013), 891-908.  doi: 10.3934/dcdss.2013.6.891. [10] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.  doi: 10.1007/s00033-011-0166-8. [11] S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.  doi: 10.1017/S0308210509000584. [12] W. Dai, J. Huang, Y. Qin, B. Wang and Y. Fang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117. [13] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746. [14] X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potential, Commun. Pure Appl. Anal., 10 (2011), 1111-1119.  doi: 10.3934/cpaa.2011.10.1111. [15] S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460.  doi: 10.2140/apde.2011.4.405. [16] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.  doi: 10.1007/s00526-006-0013-5. [17] Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.  doi: 10.1007/s00209-012-1036-6. [18] Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282. [19] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.  doi: 10.3934/dcds.2016.36.3277. [20] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7. [21] C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.  doi: 10.1137/080712301. [22] C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115. [23] Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.  doi: 10.4171/JEMS/6. [24] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293. [25] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032. [26] E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Rhode Island, 2001. [27] E. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845. [28] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [29] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020. [30] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949.  doi: 10.1016/j.jmaa.2007.12.064. [31] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3. [32] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl., 91 (2009), 49-79.  doi: 10.1016/j.matpur.2008.09.003. [33] V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differential Equations, 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019. [34] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007. [35] X. Shang and J. Zhang, Multi-peak positive solutions for a fractional nonlinear elliptic equation, Discrete Contin. Dyn. Syst., 35 (2015), 3183-3201.  doi: 10.3934/dcds.2015.35.3183. [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. doi: 10.1007/s00033-017-0806-8. [37] E. Stein, Singular Integrals and Differentiability Properties of Function, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. [38] S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263 (2012), 3857-3882.  doi: 10.1016/j.jfa.2012.09.012. [39] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.  doi: 10.1007/BF01208265. [40] W. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math. Vol. 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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##### References:
 [1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443.  doi: 10.1007/s00209-004-0663-y. [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. doi: 10.1090/cln/010. [3] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010. [4] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167. [5] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038. [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. doi: 10.1007/s00526-017-1110-3. [7] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [8] Y. Chen and H. Gao, The Cauchy problem for the Hartree equations under random influences, J. Differential Equations, 259 (2015), 5192-5219.  doi: 10.1016/j.jde.2015.06.021. [9] S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. S., 6 (2013), 891-908.  doi: 10.3934/dcdss.2013.6.891. [10] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.  doi: 10.1007/s00033-011-0166-8. [11] S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.  doi: 10.1017/S0308210509000584. [12] W. Dai, J. Huang, Y. Qin, B. Wang and Y. Fang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117. [13] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746. [14] X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potential, Commun. Pure Appl. Anal., 10 (2011), 1111-1119.  doi: 10.3934/cpaa.2011.10.1111. [15] S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460.  doi: 10.2140/apde.2011.4.405. [16] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.  doi: 10.1007/s00526-006-0013-5. [17] Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.  doi: 10.1007/s00209-012-1036-6. [18] Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282. [19] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.  doi: 10.3934/dcds.2016.36.3277. [20] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7. [21] C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.  doi: 10.1137/080712301. [22] C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115. [23] Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.  doi: 10.4171/JEMS/6. [24] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293. [25] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032. [26] E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Rhode Island, 2001. [27] E. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845. [28] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [29] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020. [30] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949.  doi: 10.1016/j.jmaa.2007.12.064. [31] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3. [32] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl., 91 (2009), 49-79.  doi: 10.1016/j.matpur.2008.09.003. [33] V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differential Equations, 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019. [34] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007. [35] X. Shang and J. Zhang, Multi-peak positive solutions for a fractional nonlinear elliptic equation, Discrete Contin. Dyn. Syst., 35 (2015), 3183-3201.  doi: 10.3934/dcds.2015.35.3183. [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. doi: 10.1007/s00033-017-0806-8. [37] E. Stein, Singular Integrals and Differentiability Properties of Function, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. [38] S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263 (2012), 3857-3882.  doi: 10.1016/j.jfa.2012.09.012. [39] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.  doi: 10.1007/BF01208265. [40] W. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math. Vol. 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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