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

Wei Dai; Jiahui Huang; Yu Qin; Bo Wang; Yanqin Fang

doi:10.3934/dcds.2018117

Previous Article
Regularity estimates for nonlocal Schrödinger equations
DCDS Home
This Issue
Next Article
Non-existence of positive solutions for a higher order fractional equation

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

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

Wei Dai ^1,, , Jiahui Huang ^1, , Yu Qin ^1, , Bo Wang ^1, and Yanqin Fang ^2,3,

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).

Keywords: Fractional Laplacians, positive solutions, radial symmetry, uniqueness, regularity, Hartree type nonlinearity, methods of moving planes in integral forms.

Mathematics Subject Classification: Primary: 35R11, 35J91; Secondary: 35B06, 35B65.

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:

[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. 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. 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. 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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. 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. Gidas, W. 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. 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. 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. 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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. 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. Gidas, W. 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. 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. 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. 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. 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. 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. Google Scholar
[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. 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

[1]	Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083
[2]	Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685
[3]	Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051
[4]	Zongming Guo, Xuefei Bai. On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1091-1107. doi: 10.3934/cpaa.2008.7.1091
[5]	Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121
[6]	Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113
[7]	Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122
[8]	Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015
[9]	Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201
[10]	Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[11]	Yonggeun Cho, Tohru Ozawa. On radial solutions of semi-relativistic Hartree equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 71-82. doi: 10.3934/dcdss.2008.1.71
[12]	Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020034
[13]	Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102
[14]	Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[15]	Cristian Bereanu, Petru Jebelean, Jean Mawhin. Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 637-648. doi: 10.3934/dcds.2010.28.637
[16]	Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[17]	Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531
[18]	Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.
[19]	Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475
[20]	Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors