Existence and asymptotical behavior of the minimizer of Hartree type equation with periodic potentials

References

• Lichtenstein L. Gleichgewichtsfiguren der rotierenden Plüssigkeiten. Berlin: Springer; 1933.
   Google Scholar
• Dalfovo F, Giorgini S, Pitaevskii L-P, et al. Theory of Bose-Einstein condensation in trapped gases. Rev Mod Phys. 1999;71:463–512. doi: 10.1103/RevModPhys.71.463
   Web of Science ®Google Scholar
• Pitaevslii L-P. Vortex lines in an imperfect Bose gas. Sov Phys JETP. 1961;13:451–454.
   Google Scholar
• Guo Y-J, Seiringer R. On the mass concentration for Bose-Einstein condensation with attractive interactions. Lett Math Phys. 2014;104:141–156. doi: 10.1007/s11005-013-0667-9
   Web of Science ®Google Scholar
• Guo Y-J, Wang Z-Q, Zeng X-Y, et al. Properties for ground states of attractive Gross-Pitaevskii equations with multi-well potentials. Nonlinearity. 2018;31:957–979. doi: 10.1088/1361-6544/aa99a8
   Web of Science ®Google Scholar
• Guo Y-J, Zeng X-Y, Zhou H-S. Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials. Ann Inst H Poincaré Anal Non Linaire. 2016;33:809–828. doi: 10.1016/j.anihpc.2015.01.005
   Web of Science ®Google Scholar
• Auchmuty JFG. Existence of axisymmetric equilibrium figures. Arch Ration Mech Anal. 1977;65:249–261. doi: 10.1007/BF00280443
   Web of Science ®Google Scholar
• Auchmuty JFG, Beals R. Variational solutions of some nonlinear free boundary problems. Arch Ration Mech Anal. 1971;43:255–271. doi: 10.1007/BF00250465
   Web of Science ®Google Scholar
• Auchmuty JFG, Beals R. Models of rotating stars. Astrophys J. 1971;165:79–82. doi: 10.1086/180721
   Web of Science ®Google Scholar
• Caffarelli L, Friedman A. The shape of axisymmetric rotating fluid. J Funct Anal. 1980;35:109–142. doi: 10.1016/0022-1236(80)90082-8
   Web of Science ®Google Scholar
• Friedman A, Turkington B. The oblateness of an axisymmetric rotating fluid. Indiana Univ Math J. 1980;29:777–792. doi: 10.1512/iumj.1980.29.29056
   Web of Science ®Google Scholar
• Lions P-L. Minimization problems in L1(R3). J Funct Anal. 1981;41:236–275. doi: 10.1016/0022-1236(81)90089-6
   Google Scholar
• Lions PL. Solutions of Hartree-Fock equations for Coulomb systems. Commun Math Phys. 1987;109:33–97. doi: 10.1007/BF01205672
   Web of Science ®Google Scholar
• Cazenave T, Lions P-L. Oribital stability of standing waves for some nonlinear Schrödinger equations. Commun Math Phys. 1982;85:549–561. doi: 10.1007/BF01403504
   Web of Science ®Google Scholar
• Lions P-L. The concentration-compactness principle in the calculus of variations The locally compact case: Part 1. Ann Inst Henri Poincare. 1984;1:109–145. Part 2, 223–283. doi: 10.1016/S0294-1449(16)30428-0
   Web of Science ®Google Scholar
• Battaglia L, Van Schaftingen J. Groundstates of the Choquard equations with a sign-changing self-interaction potential. Z Angew Math Phys. 2018;69:69–86. doi: 10.1007/s00033-018-0975-0
   Web of Science ®Google Scholar
• Cassani D, Van Schaftingen J, Zhang J-J. Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent, to appear in Proc. Roy. Soc. Edinb. A.
   Google Scholar
• Cingolani S, Clapp M, Secchi S. Multiple solutions to a magnetic nonlinear Choquard equation. Z Angew Math Phys. 2012;63:233–248. doi: 10.1007/s00033-011-0166-8
   Web of Science ®Google Scholar
• Cingolani S, Secchi S. Multiple S1-orbits for the Schrödinger-Newton system. Diff Integral Equ. 2013;26:867–884.
   Web of Science ®Google Scholar
• Ghimenti M, Moroz V, Van Schaftingen J. Least action nodal solutions for the quadratic Choquard equation. Proc Am Math Soc. 2017;145:737–747. doi: 10.1090/proc/13247
   Web of Science ®Google Scholar
• Georgiev V, Venkov G. Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential. J Differ Equ. 2011;251(2):420–438. doi: 10.1016/j.jde.2011.04.012
   Web of Science ®Google Scholar
• Ghimenti M, Van Schaftingen J. Nodal solutions for the Choquard equation. J Funct Anal. 2016;271(1):107–135. doi: 10.1016/j.jfa.2016.04.019
   Web of Science ®Google Scholar
• Lieb E-H, Simon B. The Hartree-Fock theory for Coulomb systems. Comm Math Phys. 1977;53(3):185–194. doi: 10.1007/BF01609845
   Web of Science ®Google Scholar
• Lions P-L. Some remarks on Hartree equation. Nonlinear Anal. 1981;5(11):1245–1256. doi: 10.1016/0362-546X(81)90016-X
   Web of Science ®Google Scholar
• Moroz V, Van Schaftingen J. Existence of groundstates for a class of nonlinear Choquard equations. Trans Am Math Soc. 2015;367(9):6557–6579. doi: 10.1090/S0002-9947-2014-06289-2
   Web of Science ®Google Scholar
• Moroz V, Van Schaftingen J. Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent. Commun Contemp Math. 2015;17(5):1550005. 12. doi: 10.1142/S0219199715500054
   Web of Science ®Google Scholar
• Ruiz D, Van Schaftingen J. Odd symmetry of least energy nodal solutions for the Choquard equation. J Diff Equ. 2018;264:1231–1262. doi: 10.1016/j.jde.2017.09.034
   Web of Science ®Google Scholar
• Cingolani S, Secchi S, Squassina M. Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc Roy Soc Edinburgh Sect A. 2010;140:973–1009. doi: 10.1017/S0308210509000584
   Google Scholar
• Moroz V, Van Schaftingen J. Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J Funct Anal. 2013;265(2):153–184. doi: 10.1016/j.jfa.2013.04.007
   Web of Science ®Google Scholar
• Van Schaftingen J, Xia J-K. Standing waves with a critical frequency for nonlinear Choquard equations. Nonlinear Anal. 2017;161:87–107. doi: 10.1016/j.na.2017.05.014
   Web of Science ®Google Scholar
• Devreese J-T, Alexandrov A-S. Advances in polaron physics. Berlin: Springer; 2010. (Springer Series in Solid-State Sciences; vol. 159).
   Google Scholar
• Pekar S. Untersuchung über die Elektronentheorie der Kristalle. Berlin: Akademie Verlag; 1954.
   Google Scholar
• Lieb E-H. Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies Appl Math. 1976;57(2):93–105. doi: 10.1002/sapm197757293
   Web of Science ®Google Scholar
• Jones KRW. Newtonian quantum gravity. Aust J Phys. 1995;48:1055–1081. doi: 10.1071/PH951055
   Google Scholar
• Moroz IM, Penrose R, Tod P. Spherically-symmetric solutions of the Schrödinger-Newton equations. Classical Quantum Gravity. 1998;15:2733–2742. doi: 10.1088/0264-9381/15/9/019
   Web of Science ®Google Scholar
• Ma L, Zhao L. Classification of positive solitary solutions of the nonlinear Choquard equation. Arch Ration Mech Anal. 2010;195:455–467. doi: 10.1007/s00205-008-0208-3
   Web of Science ®Google Scholar
• Moroz V, Van Schaftingen J. A guide to the Choquard equation. J Fixed Point Theory Appl. 2017;19:773–813. doi: 10.1007/s11784-016-0373-1
   Web of Science ®Google Scholar
• Deng Y-B, Lu L, Shuai W. Constraint minimizers of mass critical Hartree energy functionals: existence and mass concentration. J Math Phys. 2015;56:061503. doi: 10.1063/1.4922368
   Web of Science ®Google Scholar
• Wang Q-X, Zhao D. Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials. J. Diff Equ. 2017;262:2684–2704. doi: 10.1016/j.jde.2016.11.004
   Web of Science ®Google Scholar
• Cao DM, Su Y-M. Minimal blow up solutions of mass-critical inhomogeneous Hartree equation. J Math Phys. 2013;54:121511. doi: 10.1063/1.4850879
   Web of Science ®Google Scholar