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We study the nonuniformly elliptic, nonlinear system $$displaylines{ - hbox{div}(h_1(x) abla u)+ a(x)u = f(x,u,v) quad ext{in } mathbb{R}^N,cr - hbox{div}(h_2(x) abla v)+ b(x)v = g(x,u,v) quad ext{in } mathbb{R}^N. }$$ Under growth and regularity conditions on the nonlinearities $f$ and $g$, we obtain weak solutions in a subspace of the Sobolev space $H^1(mathbb{R}^N, mathbb{R}^2)$ by applying a variant of the Mountain Pass Theorem.