Semi-classical Analysis Around Local Maxima and Saddle Points for Degenerate Nonlinear Choquard Equations

Abstract

We study existence of semi-classical states for the nonlinear Choquard equation: -ε2Δv+V(x)v=1εα(Iα∗F(v))f(v)inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\varepsilon ^2\\Delta v+ V(x)v = {1\\over \\varepsilon ^\\alpha }(I_\\alpha *F(v))f(v) \\quad \ ext {in}\\ {\\mathbb {R}}^N, \\end{aligned}$$\\end{document}where Nge 3, alpha in (0,N), I_alpha (x)=A_alpha /|{x}|^{N-alpha } is the Riesz potential, Fin C^1({mathbb {R}},{mathbb {R}}), F'(s)=f(s) and varepsilon >0 is a small parameter. We develop a new variational approach, in which our deformation flow is generated through a flow in an augmented space to get a suitable compactness property and to reflect the properties of the potential. Furthermore our flow keeps the size of the tails of the function small and it enables us to find a critical point without introducing a penalization term. We show the existence of a family of solutions concentrating to a local maximum or a saddle point of V(x)in C^N({mathbb {R}}^N,{mathbb {R}}) under general conditions on F(s). Our results extend the results by Moroz and Van Schaftingen (Calc Var Partial Differ Equ 52:199–235, 2015) for local minima (see also Cingolani and Tanaka (Rev Mat Iberoam 35(6):1885–1924, 2019)) and Wei and Winter (J Math Phys 50:012905, 2009) for non-degenerate critical points of the potential.