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 & Continuous Dynamical Systems - A, 2019, 39 (3) : 1389-1403. doi: 10.3934/dcds.2018117
References:
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J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996.  Google Scholar

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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.  Google Scholar

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L. CaffarelliB. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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W. ChenY. 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.  Google Scholar

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

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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

[11]

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

[12]

W. ChenC. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[15]

W. DaiZ. 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.  Google Scholar

[16]

W. DaiZ. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

[21]

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

[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.  Google Scholar

[23]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar

[24]

D. LiC. 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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

C. MiaoG. 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.  Google Scholar

[36]

C. MiaoG. 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.  Google Scholar

[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.  Google Scholar

[38]

B. Ou, A Remark on a singular integral equation, Houston J. Math., 25 (1999), 181-184.   Google Scholar

[39]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.   Google Scholar

[40]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. CaffarelliB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

W. ChenY. 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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

W. ChenC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. DaiZ. 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.  Google Scholar

[16]

W. DaiZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar

[24]

D. LiC. 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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

C. MiaoG. 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.  Google Scholar

[36]

C. MiaoG. 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.  Google Scholar

[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.  Google Scholar

[38]

B. Ou, A Remark on a singular integral equation, Houston J. Math., 25 (1999), 181-184.   Google Scholar

[39]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.   Google Scholar

[40]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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