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Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities. (English) Zbl 1485.35229

Summary: We prove existence of infinitely many solutions \(u \in H^1_r(\mathbb{R}^N)\) for the nonlinear Choquard equation \[ -\varDelta u + \mu u =(I_\alpha\ast F(u)) f(u) \quad \text{in } \mathbb{R}^N, \] where \(N\ge 3\), \(\alpha \in (0,N)\), \(I_\alpha(x) := \frac{\varGamma(\frac{N-\alpha}{2})}{\varGamma(\frac{\alpha}{2})\pi^{N/2} 2^\alpha} \frac{1}{|x|^{N-\alpha}}\), \(x \in\mathbb{R}^N \setminus \{0\}\) is the Riesz potential, and \(F\) is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: \(\mu\) is a fixed positive constant or \(\mu\) is unknown and the \(L^2\)-norm of the solution is prescribed, i.e. \(\int_{\mathbb{R}^N} |u|^2 =m>0\). Since the presence of the nonlocality prevents to apply the classical approach, introduced by H. Berestycki and P.-L. Lions [Arch. Ration. Mech. Anal. 82, 347–375 (1983; Zbl 0556.35046)], 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 V. Moroz and J. Van Schaftingen [Trans. Am. Math. Soc. 367, No. 9, 6557–6579 (2015; Zbl 1325.35052)].

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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