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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|^α}\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$,$ $N\geq3$, $α\in(0,N)$ and $2-(α/N) <p<(2N-α)/(N-2)$. \ 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 \[ u(gx)=τ(g)u(x)\text{\ \ \ for all}g\in G,\text{}x\in\mathbb{R}^{N}, \] where $τ:G\rightarrow\mathbb{S}^{1}$ is a given group homomorphism into the unit complex numbers.