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Applicable Analysis
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Volume 100, 2021 - Issue 1
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Articles

Multiplicity and concentration behavior of solutions of the critical Choquard equation

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Pages 167-190 | Received 10 Sep 2018, Accepted 15 Mar 2019, Published online: 02 Apr 2019

References

  • Pekar S. Untersuchung über die Elektronentheorie der Kristalle. Berlin: Akademie; 1954.
  • Lieb EH, Simon B. The Hartree-Fock theory for Coulomb systems. Comm Math Phys. 1977;53:185–194. doi: 10.1007/BF01609845
  • Moroz IM, Penrose R, Tod P. Spherically-symmetric solutions of the Schröinger-Newton equations. Classical Quant Grav. 1998;15:2733–2742. doi: 10.1088/0264-9381/15/9/019
  • Lieb EH. Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud Appl Math. 1976/1977;57:93–105. doi: 10.1002/sapm197757293
  • Lions PL. The Choquard equation and related questions. Nonlinear Anal. 1980;4:1063–1072. doi: 10.1016/0362-546X(80)90016-4
  • Ma L, Zhao L. Classification of positive solitary solutions of the nonlinear Choquard equation. Arch Rational Mech An. 2010;195:455–467. doi: 10.1007/s00205-008-0208-3
  • Moroz V, Van Schaftingen J. Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J Funct Anal. 2013;265:153–184. doi: 10.1016/j.jfa.2013.04.007
  • Moroz V, Van Schaftingen J. Existence of groundstates for a class of nonlinear Choquard equations. T Am Math Soc. 2015;367:6557–6579. doi: 10.1090/S0002-9947-2014-06289-2
  • Alves CO, Nóbrega AB, Yang M. Multi-bump solutions for Choquard equation with deepening potential well. Calc Var Partial Dif. 55(2016):48.
  • 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
  • Ghimenti M, Van Schaftingen J. Nodal solutions for the Choquard equation. J Funct Anal. 2016;271:107–135. doi: 10.1016/j.jfa.2016.04.019
  • Gao F, Yang M. On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation. Sci China Math. 2018;61:1219–1242. doi: 10.1007/s11425-016-9067-5
  • Moroz V, Van Schaftingen J. Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains. J Differ Equ. 2013;254:3089–3145. doi: 10.1016/j.jde.2012.12.019
  • Moroz V, Van Schaftingen J. Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent. Commun Contemp Math. 2015;17:1550005. doi: 10.1142/S0219199715500054
  • Moroz V, Van Schaftingen J. A guide to the Choquard equation. J Fix Point Theory A. 2017;19:773–813. doi: 10.1007/s11784-016-0373-1
  • Rabinowitz PH. On a class of nonlinear Schrödinger equations. Z Angew Math Phys. 1992;43:270–291. doi: 10.1007/BF00946631
  • Wang X. On concentration of positive bound states of nonlinear Schrödinger equations. Commun Math Phys. 1993;153:229–244. doi: 10.1007/BF02096642
  • del Pino M, Felmer P. Local mountain pass for semilinear elliptic problems in unbounded domains. Calc Var Partial Dif. 1996;4:121–137. doi: 10.1007/BF01189950
  • Byeon J, Jeanjean L. Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch Rational Mech An. 2007;185:185–200. doi: 10.1007/s00205-006-0019-3
  • Jeanjean L, Tanaka K. Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc Var Partial Dif. 2004;21:287–318.
  • Cingolani S, Lazzo M. Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J Differ Equ. 2000;160:118–138. doi: 10.1006/jdeq.1999.3662
  • Ambrosetti A, Badiale M, Cingolani S. Semiclassical states of nonlinear Schrödinger equations. Arch Ration Mech An. 1997;140:285–300. doi: 10.1007/s002050050067
  • Ambrosetti A, Badiale M, Cingolani S. Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch Rational Mech An. 2001;159:253–271. doi: 10.1007/s002050100152
  • del Pino M, Felmer PL. Multi-peak bound states for nonlinear Schrödinger equations. Ann I H Poincare-An. 1998;15:127–149. doi: 10.1016/S0294-1449(97)89296-7
  • del Pino M, Felmer PL. Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math Ann. 2002;324:1–32. doi: 10.1007/s002080200327
  • Gui C. Existence of multi-bump solutions for nonlinear Schrödinger equations via variational methods. Commun Part Differ Equ. 1996;21:787–820. doi: 10.1080/03605309608821208
  • Secchi S. A note on Schröinger-Newton systems with decaying electric potential. Nonlinear Anal. 2010;72:3842–3856. doi: 10.1016/j.na.2010.01.021
  • Wei J, Winter M. Strongly interacting bumps for the Schröinger-Newton equations. J Math Phys. 2009;50:012905.
  • Cingolani S, Secchi S, Squassina M. Semi-classical limit for Schröinger equations with magnetic field and Hartree-type nonlinearities. P Roy Soc Edinb A. 2010;140:973–1009. doi: 10.1017/S0308210509000584
  • Moroz V, Van Schaftingen J. Semi-classical states for the Choquard equation. Calc Var Partial Dif. 2015;52:199–235. doi: 10.1007/s00526-014-0709-x
  • Alves CO, Yang M. Existence of semiclassical ground state solutions for a generalized Choquard equation. J Differ Equ. 2014;257:4133–4164. doi: 10.1016/j.jde.2014.08.004
  • Alves CO, Yang M. Multiplicity and concentration behavior of solutions for a quasilinear Choquard equation via penalization method. P Roy Soc Edinb A. 2016;146:23–58. doi: 10.1017/S0308210515000311
  • Alves CO, Gao F, Squassina M, et al. Singularly perturbed critical Choquard equations. J Differ Equ. 2017;263:3943–3988. doi: 10.1016/j.jde.2017.05.009
  • Lieb EH, Loss M. Analysis. Providence, RI: AMS; 2001. (Grad. Stud. Math.).
  • Willem M. Minimax theorems. Boston: Birkhäuser; 1996.
  • Cassani D, Zhang JJ Ground states and semiclassical states of nonlinear Choquard equations involving of Hardy-Littlewood-Sobolev critical growth, arXiv:1611.02919v1.
  • Cao D, Noussair E. Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN. Ann I H Poincare-An. 1996;13:567–588. doi: 10.1016/S0294-1449(16)30115-9

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