Multiple solutions to a magnetic nonlinear Choquard equation

Abstract

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$,$ $N\ge 3$, $\alpha \in (0,N)$ and $2-(\alpha /N)<p<(2N-\alpha )/(N-2)$. \ 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 $\tau :G\to {\mathbb{S}}^1$ is a given group homomorphism into the unit complex numbers.

To appear on ZAMP

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