Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation [(-$\mathrm{i}$\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^{\alpha}}\ast |u|^{p}) |u|^{p-2}u,\quad x\in$\mathbb{R}$^{N}%] where A\is a real valued vector potential, $V$ is a real valued scalar potential$,$ $N\ge 3$, $\alpha \in (0,N)$ and $2-(\alpha \mathrm{/}N)<p<(2N-\alpha )\mathrm{/}(N-2)$. \ We assume that both $A$ and $V$ are compatible with the action of some group $G$ of linear isometries of ${\mathbb{R}}^N$. We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition $$u(gx)=\tau (g)u(x)\text{ for all}g\in G,x\in {\mathbb{R}}^N,$$ where $\tau :G\to {\mathbb{S}}^1$ is a given group homomorphism into the unit complex numbers.