This paper analyzes the microwave gas discharge within elliptical waveguides excited by the fundamental mode. The Rayleigh-Ritz method has been applied to solve the continuity equation. The eigenvalue problem defined by the breakdown condition has been solved and the effective diffusion length of the elliptical waveguide has been calculated, what is used to find the corona threshold. This paper extends the microwave breakdown model developed for circular waveguides and shows the better corona withstanding capabilities of elliptical waveguides. The corona breakdown electric field threshold obtained with the variational method has been compared with the one calculated with the Finite Elements Method, showing excellent agreement.