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: - epsilon(2) Delta v + V( x) v = 1/ epsilon(alpha) ( I-alpha * F( v)) f (v) in R-N, where N >= 3, alpha is an element of (0, N), I-alpha(x) = A(alpha)/| x|(N- alpha) is the Riesz potential, F is an element of C-1( R, R), F' (s) = f (s) and epsilon > 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) is an element of C-N (R-N, 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.