Standing wave solutions to the Maxwell–Chern–Simons–Schrödinger equations

We prove the existence of standing wave solutions for the Maxwell–Chern–Simons–Schrödinger equation. This model describes the fractional quantum Hall effect and anyonic superconductivity, but standing wave solutions could not be constructed by the standard arguments due to the critical singularity in the gauge fields. To overcome this difficulty, we introduce a perturbation argument by means of the Chern–Simons term. Then our result is established by applying a limiting argument and the mountain pass theorem to a functional that is obtained by representing the gauge fields in terms of a scalar field.