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We deal with the existence of constant sign solutions for the following variable exponent system Neumann boundary value problem: - div ( | ∇ u | p ( x ) - 2 ∇ u ) + λ | u | p ( x ) - 2 u = F u ( x , u , v ) in Ω , - div ( | ∇ v | q ( x ) - 2 ∇ v ) + λ | v | q ( x ) - 2 v = F v ( x , u , v ) in Ω , ∂ u / ∂ γ = 0 = ∂ v / ∂ γ on ∂ Ω . We give several sufficient conditions for the existence of the constant sign solutions, when F ( x , · , · ) satisfies neither sub-( p ( x ) , q ( x ) ) growth condition, nor Ambrosetti-Rabinowitz condition (subcritical). In particular, we obtain the existence of eight constant sign solutions.

Constant Sign Solutions for Variable Exponent System Neumann Boundary Value Problems with Singular Coefficient