  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.
