Multiplicity and Concentration of Solutions for a Fractional Kirchhoff Equation with Magnetic Field and Critical Growth

Abstract

We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth:

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

where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta )^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb {R}^{3}\rightarrow \mathbb {R}^{3}$ is a smooth magnetic potential, $V:\mathbb {R}^{3}\rightarrow \mathbb {R}$ is a positive continuous potential verifying the global condition due to Rabinowitz (Z Angew Math Phys 43:270–291, 1992), and $f:\mathbb {R}\rightarrow \mathbb {R}$ is a $C^{1}$ subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions (Ann Inst H Poincaré Anal Non Linéaire 1(2):109–145, 1984), a fractional Kato’s type inequality and the Ljusternik–Schnirelmann theory of critical points.