Vortex equations governing the fractional quantum Hall effect

An existence theory is established for a coupled non-linear elliptic system, known as "vortex equations", describing the fractional quantum Hall effect in 2-dimensional double-layered electron systems. Via variational methods, we prove the existence and uniqueness of multiple vortices over a doubly periodic domain and the full plane. In the doubly periodic situation, explicit sufficient and necessary conditions are obtained that relate the size of the domain and the vortex numbers. For the full plane case, existence is established for all finite-energy solutions and exponential decay estimates are proved. Quantization phenomena of the magnetic flux are found in both cases.