A design of 220 GHz third harmonic photonic band gap cavity gyrotron oscillator is proposed. Higher photonic crystal band gap is used to ensure the interaction between the high order electromagnetic mode (TE63-like) and the third harmonic electron cyclotron mode in gyrotron. The weak mode competition between TE63-like and TE92-like mode is studied by using a nonlinear theory, and the working conditions that ensure the start-up of the gyrotron to work in the third harmonic mode, as well as the nonlinear characteristics during the start-up process, are achieved. These results are in good agreement with the PIC (particle-in-cell) simulation. Our study shows that by using the photonic crystal as the high-frequency structure of gyrotron, high-order electromagnetic modes can interact with harmonic electron cyclotron modes efficiently.