Abstract: In this paper, the Dirichlet problem of a class of quasilinear elliptic systems with indefinite weights, which comes from mathematical models of Non Newton fluid flow, population evolution, pattern formation etc, is discussed by using Mountain Pass Lemma in critical point theory. The existence of a nontrivial weak solution is established with the help of the first eigenvalue of the corresponding eigenvalue problem, where the eigenvalue parameter is less than a known constant.

Key words: quasilinear elliptic system, indefinite weights, Mountain Pass Lemma