# American Institute of Mathematical Sciences

March  2019, 39(3): 1389-1403. doi: 10.3934/dcds.2018117

## Regularity and classification of solutions to static Hartree equations involving fractional Laplacians

 1 School of Mathematics and Systems Science, Beihang University (BUAA), Beijing 100083, China 2 School of Mathematics, Hunan University, Changsha 410082, China 3 School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522, NSW, Australia

* Corresponding author: Wei Dai at weidai@buaa.edu.cn

Received  May 2017 Revised  November 2017 Published  April 2018

Fund Project: The first author was supported by the NNSF of China (No. 11501021), the second author was supported by the NNSF of China (No. 11301166).

In this paper, we are concerned with the fractional order equations (1) with Hartree type $\dot{H}^{\frac{α}{2}}$-critical nonlinearity and its equivalent integral equations (3). We first prove a regularity result which indicates that weak solutions are smooth (Theorem 1.2). Then, by applying the method of moving planes in integral forms, we prove that positive solutions $u$ to (1) and (3) are radially symmetric about some point $x_{0}∈\mathbb{R}^{d}$ and derive the explicit forms for $u$ (Theorem 1.3 and Corollary 1). As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities (Corollary 2).

Citation: Wei Dai, Jiahui Huang, Yu Qin, Bo Wang, Yanqin Fang. Regularity and classification of solutions to static Hartree equations involving fractional Laplacians. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1389-1403. doi: 10.3934/dcds.2018117
##### References:
 [1] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. [2] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025. [3] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903. [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304. [5] D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Royal Soc. Edinburgh-A: Math., 97 (2018), 255-273.  doi: 10.1080/00036811.2016.1260708. [6] S. A. Chang and P. C. Yang, On uniqueness of solutions of $n$-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102.  doi: 10.4310/MRL.1997.v4.n1.a9. [7] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013. [8] 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. [9] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys., Vol. 4, 2010. [10] 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. [11] 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. [12] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445. [13] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, 1-43, Lecture Notes in Math., 1871, Springer, Berlin, 2006. [14] W. Dai and Z. Liu, Classification of positive solutions to a system of Hardy-Sobolev type equations, Acta Mathematica Scientia, 37 (2017), 1415-1436.  doi: 10.1016/S0252-9602(17)30082-6. [15] W. Dai, Z. Liu and G. Lu, Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space, Potential Analysis, 46 (2017), 569-588.  doi: 10.1007/s11118-016-9594-6. [16] W. Dai, Z. Liu and G. Lu, Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space, Comm. Pure Appl. Anal., 16 (2017), 1253-1264.  doi: 10.3934/cpaa.2017061. [17] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018. [18] J. Frohlich and E. Lenzmann, Mean-field limit of quantum bose gases and nonlinear Hartree equation, in: Sminaire E. D. P. (2003-2004), Expos nXVIII, (2004), 26pp. [19] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{n}$, Mathematical Analysis and Applications, Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. [20] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125. [21] Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282. [22] 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. [23] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023. [24] D. Li, C. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Diff. Equations, 246 (2009), 1139-1163.  doi: 10.1016/j.jde.2008.05.013. [25] 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. [26] E. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845. [27] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032. [28] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^{n}$, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052. [29] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, parts1 and 2, Ann. Inst. H. Poincaré Anal. Non Linéaire., 1 (1984), 109-145 and 223--283.  doi: 10.1016/S0294-1449(16)30422-X. [30] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, parts1 and 2, Revista Math. Iberoamericana, 1 (1985), 145-201 and 45--121. [31] Z. Liu and W. Dai, A Liouville type theorem for poly-harmonic system with Dirichlet boundary conditions in a half space, Advanced Nonlinear Studies, 15 (2015), 117-134.  doi: 10.1515/ans-2015-0106. [32] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014. [33] 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. [34] 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. [35] C. Miao, G. Xu and L. Zhao, Global wellposedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Diff. Equations, 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011. [36] C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., 114 (2009), 213-236.  doi: 10.4064/cm114-2-5. [37] 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. [38] B. Ou, A Remark on a singular integral equation, Houston J. Math., 25 (1999), 181-184. [39] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. [40] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970. [41] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258. [42] D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.  doi: 10.1016/j.aml.2014.12.007.

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##### References:
 [1] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. [2] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025. [3] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903. [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304. [5] D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Royal Soc. Edinburgh-A: Math., 97 (2018), 255-273.  doi: 10.1080/00036811.2016.1260708. [6] S. A. Chang and P. C. Yang, On uniqueness of solutions of $n$-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102.  doi: 10.4310/MRL.1997.v4.n1.a9. [7] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013. [8] 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. [9] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys., Vol. 4, 2010. [10] 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. [11] 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. [12] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445. [13] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, 1-43, Lecture Notes in Math., 1871, Springer, Berlin, 2006. [14] W. Dai and Z. Liu, Classification of positive solutions to a system of Hardy-Sobolev type equations, Acta Mathematica Scientia, 37 (2017), 1415-1436.  doi: 10.1016/S0252-9602(17)30082-6. [15] W. Dai, Z. Liu and G. Lu, Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space, Potential Analysis, 46 (2017), 569-588.  doi: 10.1007/s11118-016-9594-6. [16] W. Dai, Z. Liu and G. Lu, Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space, Comm. Pure Appl. Anal., 16 (2017), 1253-1264.  doi: 10.3934/cpaa.2017061. [17] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018. [18] J. Frohlich and E. Lenzmann, Mean-field limit of quantum bose gases and nonlinear Hartree equation, in: Sminaire E. D. P. (2003-2004), Expos nXVIII, (2004), 26pp. [19] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{n}$, Mathematical Analysis and Applications, Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. [20] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125. [21] Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282. [22] 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. [23] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023. [24] D. Li, C. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Diff. Equations, 246 (2009), 1139-1163.  doi: 10.1016/j.jde.2008.05.013. [25] 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. [26] E. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845. [27] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032. [28] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^{n}$, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052. [29] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, parts1 and 2, Ann. Inst. H. Poincaré Anal. Non Linéaire., 1 (1984), 109-145 and 223--283.  doi: 10.1016/S0294-1449(16)30422-X. [30] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, parts1 and 2, Revista Math. Iberoamericana, 1 (1985), 145-201 and 45--121. [31] Z. Liu and W. Dai, A Liouville type theorem for poly-harmonic system with Dirichlet boundary conditions in a half space, Advanced Nonlinear Studies, 15 (2015), 117-134.  doi: 10.1515/ans-2015-0106. [32] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014. [33] 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. [34] 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. [35] C. Miao, G. Xu and L. Zhao, Global wellposedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Diff. Equations, 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011. [36] C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., 114 (2009), 213-236.  doi: 10.4064/cm114-2-5. [37] 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. [38] B. Ou, A Remark on a singular integral equation, Houston J. Math., 25 (1999), 181-184. [39] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. [40] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970. [41] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258. [42] D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.  doi: 10.1016/j.aml.2014.12.007.
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