On the existence of positive solutions for a class of ()-Laplacian system

We consider the system { − Δ p ( x ) u = λ 1 a ( x ) f ( v ) + μ 1 α ( x ) h ( u ) , in  Ω , − Δ q ( x ) v = λ 2 b ( x ) g ( u ) + μ 2 β ( x ) γ ( v ) , in  Ω , u = 0 = v , on  ∂ Ω , where p ( x ) ∈ C 1 ( R N ) is a radial symmetric function such that sup | ∇ p ( x ) | < ∞ , 1 < inf p ( x ) ≤ sup p ( x ) < ∞ , a , b , α , β : [ 0 , + ∞ ) → ( 0 , ∞ ) is a continuous function and Ω = B ( 0 , R ) ⊂ R N is a bounded radial symmetric domain, and where − Δ p ( x ) u = − div ( | ∇ u | p ( x ) − 2 ∇ u ) which is called the p ( x ) -Laplacian. We discuss the existence of positive solution via sub-super-solutions without assuming sign conditions on f ( 0 ) , h ( 0 ) , g ( 0 ) and γ ( 0 ) .