Multiple solutions for critical nonlocal elliptic problems with magnetic field

Abstract

In this paper, we consider the existence of multiple solutions of the following critical nonlocal elliptic equations with magnetic field: \begin{document}$\begin{align} \left\{\begin{aligned} (-i\nabla-A(x))^2u& = \lambda |u|^{p-2}u+\left(\int_{\Omega}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^{\alpha}}dy\right)|u|^{2^*_\alpha-2}u\quad {\rm in}\quad \Omega,\\ &u = 0\quad \partial\Omega,\\ \end{aligned}\right. \end{align}\;\;\;\;(1)$ \end{document} where $ i $ is imaginary unit, $ N\geq4 $, $ 2^*_\alpha = \frac{2N-\alpha}{N-2} $ with $ 0<\alpha<4 $, $ \lambda>0 $ and $ 2\leq p<2^* = \frac{2N}{N-2} $. Suppose the magnetic vector potential $ A(x) = (A_1(x), A_2(x),..., A_N(x)) $ is real and local Hölder continuous, We show by the Ljusternik-Schnirelman theory that (1) has at least $ cat_\Omega(\Omega) $ nontrivial solutions for $ \lambda $ small.