TY - JOUR

T1 - Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities

AU - Cingolani, Silvia

AU - Gallo, Marco

AU - Tanaka, Kazunaga

N1 - Funding Information:
The first and second authors are supported by MIUR-PRIN project “Qualitative and quantitative aspects of nonlinear PDEs” (2017JPCAPN_005), and partially supported by GNAMPA-INdAM. The third author is supported in part by Grant-in-Aid for Scientific Research (19H00644, 18KK0073, 17H02855, 16K13771) of Japan Society for the Promotion of Science.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/4

Y1 - 2022/4

N2 - 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).

AB - 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).

KW - Even and odd nonlinearities

KW - Lagrange multiplier

KW - Multidimensional odd paths

KW - Nonlinear Choquard equation

KW - Nonlocal source

KW - Normalized solutions

KW - Pohozaev’s identity

KW - Radially symmetric solutions

KW - Riesz potential

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U2 - 10.1007/s00526-021-02182-4

DO - 10.1007/s00526-021-02182-4

M3 - Article

AN - SCOPUS:85124480576

SN - 0944-2669

VL - 61

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 2

M1 - 68

ER -