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

Abstract

AbstractWe study existence of semi-classical states for the nonlinear Choquard equation: $$\begin{aligned} -\varepsilon ^2\Delta v+ V(x)v = {1\over \varepsilon ^\alpha }(I_\alpha *F(v))f(v) \quad \text {in}\ {\mathbb {R}}^N, \end{aligned}$$ - ε 2 Δ v + V ( x ) v = 1 ε α ( I α ∗ F ( v ) ) f ( v ) in R N , where $$N\ge 3$$ N ≥ 3 , $$\alpha \in (0,N)$$ α ∈ ( 0 , N ) , $$I_\alpha (x)=A_\alpha /|{x}|^{N-\alpha }$$ I α ( x ) = A α / | x | N - α is the Riesz potential, $$F\in C^1({\mathbb {R}},{\mathbb {R}})$$ F ∈ C 1 ( R , R ) , $$F'(s)=f(s)$$ F ′ ( s ) = f ( s ) and $$\varepsilon >0$$ ε > 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 $$V(x)\in C^N({\mathbb {R}}^N,{\mathbb {R}})$$ V ( x ) ∈ 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.