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

Abstract

AbstractWe prove existence of infinitely many solutions $$u \in H^1_r({\mathbb {R}}^N)$$ u ∈ H r 1 ( R N ) for the nonlinear Choquard equation $$\begin{aligned} - {\varDelta } u + \mu u =(I_\alpha *F(u)) f(u) \quad \hbox {in}\ {\mathbb {R}}^N, \end{aligned}$$ - Δ u + μ u = ( I α ∗ F ( u ) ) f ( u ) in R N , where $$N\ge 3$$ N ≥ 3 , $$\alpha \in (0,N)$$ α ∈ ( 0 , N ) , $$I_\alpha (x) := \frac{{\varGamma }(\frac{N-\alpha }{2})}{{\varGamma }(\frac{\alpha }{2}) \pi ^{N/2} 2^\alpha } \frac{1}{|x|^{N- \alpha }}$$ I α ( x ) : = Γ ( N - α 2 ) Γ ( α 2 ) π N / 2 2 α 1 | x | N - α , $$x \in {\mathbb {R}}^N \setminus \{0\}$$ x ∈ R N \ { 0 } is the Riesz potential, and F is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: $$\mu $$ μ is a fixed positive constant or $$\mu $$ μ is unknown and the $$L^2$$ L 2 -norm of the solution is prescribed, i.e. $$\int _{{\mathbb {R}}^N} |u|^2 =m>0$$ ∫ R N | 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).