Vortex–Type Solutions to a Magnetic Nonlinear Choquard Equation

We consider the stationary nonlinear magnetic Choquard equation
\begin{equation*}
  (-\mathrm{i}\nabla+A(x))^2u+W(x)u=\left(\frac{1}{|x|^\alpha}*|u|^p\right)
    |u|^{p-2}u,\qquad x\in\mathbb{R}^N,
\end{equation*}
where \(N\ge 3\), \(\alpha\in(0,N)\),
\(p\in\left[2,\frac{2N-\alpha}{N-2}\right)\),
\(A\colon\mathbb{R}^N\to\mathbb{R}^N\) is a magnetic potential and
\(W\colon\mathbb{R}^N\to\mathbb{R}\) is a bounded electric potential. We
assume that both \(A\) and \(W\) are compatible with the action of some
group \(\Gamma\) of linear isometries of \(\mathbb{R}^N\).

We shall give an overview and present some recent results concerning the
existence of vortex-type solutions to this equation which satisfy the symmetry
condition
\begin{equation*}
  u(\gamma x)=\phi(\gamma)u(x)
  \qquad\text{for all } \gamma\in\Gamma,\ x\in\mathbb{R}^N,
\end{equation*}
where \(\phi\colon\Gamma\to\mathbb{S}^1\) is a given continuous group
homomorphism into the unit complex numbers.