Abstract: In this paper, the solution of a class of system of nonlinear elliptic equations is transformed into a problem of finding critical points of the given functional. Under various conditions, the existence of critical points and nontrival critical points is proved by using variational methods, the extreme theory and the Mountain Pass lemma, so that the solution and nontrivial solution of the nonlinear elliptic system is obtained.

Key words: extreme theory, Mountain Pass lemma, nonlinear elliptic system, nontrivial solutions