Both experimentally and theoretically, the study of two-dimensional electron systems in strong magnetic fields is one of the most interesting topics in condensed-matter physics. In the presence of strong magnetic fields and low temperature the fractional quantum Hall effect (FQHE) can be observed in two-dimensional electron systems. Through topological excitations in Chern-Simons gauge theories, Ichinose and Sekiguchi in [26, 27] proposed a Lagrangian modelling the FQHE in a double-layer electron system. We derive the equations of motion corresponding to the proposed Lagrangian. A ''first integral'' of the equations of motion is then obtained. From the resulting ''first integral'' or self-dual equations we obtain an interesting two-dimensional non-linear coupled elliptic system. Necessary and sufficient conditions for the existence of a solution over a doubly periodic domain is then established.

Although not immediately obvious a variational principle is achieved. Therefore, the two-dimensional non-linear coupled elliptic system can be viewed as the Euler-Lagrange equations of a corresponding energy functional. We prove the existence of a minimizer of the energy functional and hence the existence of a solution of the elliptic system in two ways: via a direct minimization and a constrained minimization problem. In both approaches, we use a weak compactness argument since we will be working over reflexive Banach spaces. The main analytic tools necessary in this pursuit are Jensen's inequality, the Poincare inquality, the Trudinger-Moser inequality, and the usage of the weak lower semi-continuity of the functional. Uniqueness of the solutions follows from the convexity of the energy functional.

Solutions to the elliptic system are also studied over the full plane satisfying topological boundary conditions at infinity. A variational principle is established similarly to the doubly periodic case. We prove the existence over the full plane using a weak compactness argument. However, the full plane case is much more challenging than the doubly periodic case. The main difficulty is seen in establishing a coercive lower bound for our energy functional. Uniqueness of the solutions again follows from a convexity argument. Through several Sobolev embedding inequalities and the maximum principle we perform an asymptotic analysis of the solutions. We establish exponential decay estimates of both the solution and their gradients. The significance of the mathematical theory established is reflected upon the simple and explicit quantization formulas describing the magnetic fluxes in a doubly periodic domain and over the full plane.