关键词: Fractional Kirchhoff-Choquard problem, penalization method, Ljusternik-Schnirelmann theory, variational methods

Abstract: In this paper, we study the multiplicity and concentration of positive solutions for the following fractional Kirchhoff-Choquard equation with magnetic fields: \begin{equation*} (a\varepsilon^{2s}+b\varepsilon^{4s-3}[u]^2_{\varepsilon,A/\varepsilon}) (-\Delta)_{A/\varepsilon}^{s} u+V(x)u = \varepsilon^{-\alpha}(I_\alpha*F(|u|^2))f(|u|^2)u\ \ \text{in }\ \mathbb{R}^3. \end{equation*} Here $\varepsilon > 0$ is a small parameter, $a,b > 0$ are constants, $s \in (0% \frac{3} {4} ,1), (-\Delta)_{A}^{s}$ is the fractional magnetic Laplacian, $A: \mathbb{R}^3 \to \mathbb{R}^3$ is a smooth magnetic potential, $I_{\alpha}=\frac{\Gamma(\frac{3-\alpha}{2})}{2^{\alpha}\pi^{\frac{3}{2}}\Gamma(\frac{\alpha}{2})}\cdot\frac{1}{|x|^{\alpha} }$ is the Riesz potential, the potential $V$ is a positive continuous function having a local minimum, and $f: \mathbb{R} \to \mathbb{R}$ is a $C^1$ subcritical nonlinearity. Under some proper assumptions regarding $V$ and $f, $ we show the multiplicity and concentration of positive solutions with the topology of the set $M:= \{x \in \mathbb{R}^3 : V (x) = \inf V \}$ by applying the penalization method and Ljusternik-Schnirelmann theory for the above equation.

Key words: Fractional Kirchhoff-Choquard problem, penalization method, Ljusternik-Schnirelmann theory, variational methods