Vortex-type solutions to a magnetic nonlinear Choquard equation

Abstract

We consider the stationary nonlinear magnetic Choquard equation

$$(-{\rm i}\nabla + A(x))^{2} u + W(x)u = \left(\frac{1}{|x|^{\alpha}} \ast |u|^{p}\right) |u|^{p - 2}u, \quad x \in \mathbb{R}^N,$$

where ${N \geq 3, \alpha \in (0, N), p \in \bigl[2, \frac{2N - \alpha}{N - 2}\bigr), A : \mathbb{R}^N \rightarrow \mathbb{R}^N}$ is a magnetic potential and ${W : \mathbb{R}^N \rightarrow \mathbb{R}}$ is a bounded electric potential. For a given group ${\Gamma}$ of linear isometries of ${\mathbb{R}^N}$, we assume that A(gx) = gA(x) and W(gx) = W(x) for all ${g \in \Gamma, x \in \mathbb{R}^N}$. Under some assumptions on the decay of A and W at infinity, we establish the existence of solutions to this problem which satisfy

$$u(\gamma{x}) = \phi(\gamma)u(x) \quad {\rm for\, all} \gamma \in \Gamma,\, x \in \mathbb{R}^N,$$

where ${\phi : \Gamma \rightarrow \mathbb{S}^1}$ is a given continuous group homomorphism into the unit complex numbers.