Multiplicity and Concentration of Solutions for a Fractional Magnetic Kirchhoff Equation with Competing Potentials

Abstract

This paper is concerned with the following fractional electromagnetic Kirchhoff equation with competing potentials and critical nonlinearity

$$\begin{aligned} \left( a\varepsilon ^{2s}+b\varepsilon ^{4s-3} [u]_{A/\varepsilon }^{2}\right) (-\Delta )_{A/\varepsilon }^{s}u+V(x)u=f(|u|^{2})u+K(x)|u|^{2^{*}_{s}-2}u \quad \text{ in }\; {\mathbb {R}}^{3}, \end{aligned}$$

where $\varepsilon >0$ is a small parameter, $A\in C^{0,\alpha }({\mathbb {R}}^{3},{\mathbb {R}}^{3} )$ with exponent $\alpha \in (0,1]$, $(-\Delta )_{A/\varepsilon }^s$ is the fractional magnetic operator with $s\in (\frac{3}{4}, 1)$, $2_{s}^{*} =\frac{6}{3-2s}$ is the fractional critical exponent, and $a, b>0$ are fixed constants. Assuming that V, K and f satisfy some suitable conditions, we establish the multiplicity and concentration of solutions by variational methods and Ljusternik–Schnirelmann theory.