Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth (1810.04561v2)
Abstract: We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \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|{\2-2}u \quad \mbox{ in } \mathbb{R}{3}, \end{equation*} 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 \cite{Rab}, 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 \cite{Lions}, a fractional Kato's type inequality and the Ljusternik-Schnirelmann theory of critical points.