Semiclassical Analysis for Fractional Choquard Equations with General Nonlinearities: Multiplicity and Concentration

Abstract In this paper, we investigate the following fractional nonlinear Choquard equation: $$\begin{aligned} \left\{ \begin{array}{ll} {{\,\mathrm{\varepsilon }\,}}^{2s} (-\Delta )^{s} v +V(x) v= {{\,\mathrm{\varepsilon }\,}}^{-\alpha } (I_{\alpha }*F(v)) F'(v) \text{ in } \mathbb {R}^{N},\\ v\in H^{s}(\mathbb {R}^{N}), \,\, v>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$ where $${{\,\mathrm{\varepsilon }\,}}>0$$ is a small parameter, $$s\in (0, 1)$$ , $$N\ge 2$$ , $$(-\Delta )^{s}$$ denotes the fractional Laplacian, and $$I_{\alpha }$$ is the Riesz potential of order $$\alpha \in ((N-4s)_{+}, N)$$ . The potential $$V\in C^{0}(\mathbb {R}^N, (0, +\infty ))$$ satisfies $$\begin{aligned} m_{0}:=\inf _{\Omega }V<\min _{\partial \Omega }V, \end{aligned}$$ for some bounded open set $$\Omega \subset \mathbb {R}^N$$ . The function $$F\in C^{1}(\mathbb {R})$$ is a nonlinearity of Berestycki–Lions type. By employing suitable variational methods, we establish the existence of at least $$\textrm{cupl}(K)+1$$ solutions concentrating around the set $$K:=\{x\in \Omega : V(x)=m_{0} \}$$ as $${{\,\mathrm{\varepsilon }\,}}\rightarrow 0^{+}.$$