Abstract: Using a threecriticalpoint theorem in variational methods, we investigated the Dirichlet problem of a class of quasilinear elliptic systems, and proved the existence of at least three weak solutions under some novel assumptions and gave, as the application of the main theorem, an example about the coupling systems come from the second harmonic waves in nonlinear optics.

Key words: threecritical-point theorem, quasilinear elliptic system, variational method