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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[3]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010. Google Scholar

[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. Google Scholar

[5]

W. ChenC. 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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[9]

S. CingolaniM. 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. Google Scholar

[10]

S. CingolaniM. 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. Google Scholar

[11]

S. CingolaniS. 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. Google Scholar

[12]

W. DaiJ. HuangY. QinB. 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. Google Scholar

[13]

P. FelmerA. 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. Google Scholar

[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. Google Scholar

[15]

S. IbrahimN. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406. doi: 10.1137/120879282. Google Scholar

[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. Google Scholar

[20]

Y. LeiC. 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. Google Scholar

[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. Google Scholar

[22]

C. LiZ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[26]

E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Rhode Island, 2001. Google Scholar

[27]

E. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194. doi: 10.1007/BF01609845. Google Scholar

[28]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar

[29]

C. MaW. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

C. MiaoG. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

E. Stein, Singular Integrals and Differentiability Properties of Function, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. Google Scholar

[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. Google Scholar

[39]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010. Google Scholar

[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. Google Scholar

[5]

W. ChenC. 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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[9]

S. CingolaniM. 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. Google Scholar

[10]

S. CingolaniM. 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. Google Scholar

[11]

S. CingolaniS. 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. Google Scholar

[12]

W. DaiJ. HuangY. QinB. 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. Google Scholar

[13]

P. FelmerA. 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. Google Scholar

[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. Google Scholar

[15]

S. IbrahimN. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406. doi: 10.1137/120879282. Google Scholar

[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. Google Scholar

[20]

Y. LeiC. 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. Google Scholar

[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. Google Scholar

[22]

C. LiZ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[26]

E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Rhode Island, 2001. Google Scholar

[27]

E. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194. doi: 10.1007/BF01609845. Google Scholar

[28]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar

[29]

C. MaW. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

C. MiaoG. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

E. Stein, Singular Integrals and Differentiability Properties of Function, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. Google Scholar

[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. Google Scholar

[39]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265. Google Scholar

[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. Google Scholar

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