Abstract:

In this paper, we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation

$\begin{matrix}(-{\rm i}\nabla-A(x))^2u+\lambda V(x)u=\mu |u|^{p-2}u+\Big(\int_{\mathbb{R^N}}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^\alpha}{\rm d}y\Big)|u|^{2^*_\alpha-2}u\quad {\rm in}\ \mathbb{R}^N,\end{matrix}$ (0.1)

where $N\geq4$, $2\leq p<2^*$, $2^*_\alpha=\frac{2N-\alpha}{N-2}$ with $0<\alpha<4$, $\lambda>0$, $\mu\in\mathbb{R}$, $A(x)= (A_1(x), A_2(x),\cdots , A_N(x))$ is a real local Hölder continuous vector function, $i$ is the imaginary unit, and $V(x)$ is a real valued potential function on $\mathbb{R}^N$.Supposing that $\Omega={\rm int}\,V^{-1}(0)\subset\mathbb{R}^N$ is bounded, we show that problem (0.1) possesses at least cat$_\Omega(\Omega)$ nontrivial solutions if $\lambda$ is large.