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We use the sub-supersolution method and the Mountain Pass Theorem in order to show existence and multiplicity of solutions for the quasilinear system given by  − [ N ∑ i=1 ∂ ∂xi ∣∣∣∣ ∂u ∂xi ∣∣∣∣pi−2 ∂u ∂xi ] = a1(x)u + Fu(x, u, v) in Ω, − [ N ∑ i=1 ∂ ∂xi ∣∣∣∣ pi∂v ∂xi ∣∣∣∣pi−2 ∂v ∂xi ] = a2(x)v + Fv(x, u, v) in Ω, u, v > 0 in Ω, u = v = 0 on ∂Ω, where aj, j = 1, 2 are functions in L∞(Ω) and and Fu and Fv are continuous functions on Ω×R2.

Solutions to an anisotropic system via sub-supersolution method and Mountain Pass Theorem