Multiple solutions to a magnetic nonlinear Choquard equation

TL;DR

This paper proves the existence of multiple symmetric solutions to a magnetic nonlinear Choquard equation involving a vector potential, scalar potential, and nonlocal nonlinearities, under symmetry constraints and group actions.

Contribution

It introduces new methods to establish multiple solutions with symmetry properties for a class of magnetic nonlinear Choquard equations.

Findings

Multiple complex solutions satisfying symmetry conditions.

Solutions are obtained under group action and homomorphism constraints.

The results extend understanding of nonlinear Choquard equations with magnetic fields.

Abstract

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$,$ $N\geq3$, $\alpha\in(0,N)$ 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 \[ u(gx)=\tau(g)u(x)\text{\ \ \ for all}g\in G,\text{}x\in\mathbb{R}^{N}, \] where $\tau:G\rightarrow\mathbb{S}^{1}$ is a given group homomorphism into the unit complex numbers.