Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities

Silvia Cingolani*, Marco Gallo, Kazunaga Tanaka

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

We prove existence of infinitely many solutions u∈Hr1(RN) for the nonlinear Choquard equation -Δu+μu=(Iα∗F(u))f(u)inRN,where N≥ 3 , α∈ (0 , N) , Iα(x):=Γ(N-α2)Γ(α2)πN/22α1|x|N-α, x∈ RN\ { 0 } is the Riesz potential, and F is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: μ is a fixed positive constant or μ is unknown and the L2-norm of the solution is prescribed, i.e. ∫RN|u|2=m>0. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions (Arch Ration Mech Anal 82(4):347–375, 1983), we implement a new construction of multidimensional odd paths, where some estimates for the Riesz potential play an essential role, and we find a nonlocal counterpart of their multiplicity results. In particular we extend the existence results due to Moroz and Van Schaftingen (Trans Am Math Soc 367(9):6557–6579, 2015).

Original languageEnglish
Article number68
JournalCalculus of Variations and Partial Differential Equations
Volume61
Issue number2
DOIs
Publication statusPublished - 2022 Apr

Keywords

  • Even and odd nonlinearities
  • Lagrange multiplier
  • Multidimensional odd paths
  • Nonlinear Choquard equation
  • Nonlocal source
  • Normalized solutions
  • Pohozaev’s identity
  • Radially symmetric solutions
  • Riesz potential

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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