Infinite energy solutions to vortex equations governing the fractional quantum Hall effect

In this paper, we utilize weighted Sobolev spaces to establish an existence theory for infinite energy solutions to a coupled non-linear elliptic system. This system describes the fractional quantum Hall effect in two-dimensional double-layered systems. Via variational methods in a suitable weighted Sobolev space, we prove the existence of multiple vortices over the full plane. These methods include constrained minimization of an action functional and the existence of a critical point, known as a saddle point, by way of a mountain pass theorem. Furthermore, for these solutions, which are necessarily of infinite energy, we establish exponential decay estimates.