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

Abstract

We prove existence of infinitely many solutions u in H^1_r({mathbb {R}}^N) for the nonlinear Choquard equation -Δu+μu=(Iα∗F(u))f(u)inRN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} - {\\varDelta } u + \\mu u =(I_\\alpha *F(u)) f(u) \\quad \\hbox {in}\\ {\\mathbb {R}}^N, \\end{aligned}$$\\end{document}where Nge 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 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).