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

We study existence of semi-classical states for the nonlinear Choquard equation: −ε2 v + V(x)v = 1 εα (Iα ∗ F(v)) f (v) in RN , where N ≥ 3, α ∈ (0, N), Iα(x) = Aα/|x|N−α is the Riesz potential, F ∈ C1(R,R), F (s) = f (s) andε > 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) ∈ CN (RN ,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.