Existence and asymptotic behavior of positive solutions for a variable exponent elliptic system without variational structure

We mainly consider the existence and asymptotic behavior of positive solutions of the following system { − Δ p ( x ) u = λ p ( x ) ( u α ( x ) v γ ( x ) + h 1 ( x ) ) in Ω , − Δ q ( x ) v = λ q ( x ) ( u δ ( x ) v β ( x ) + h 2 ( x ) ) in Ω , u = v = 0 on ∂ Ω , where Ω ⊂ R N is a bounded domain with C 2 boundary ∂ Ω , 1 < p ( x ) , q ( x ) ∈ C 1 ( Ω ¯ ) are functions, and − Δ p ( x ) u = − div ( | ∇ u | p ( x ) − 2 ∇ u ) is called p ( x ) -Laplacian. When α , β , γ , δ satisfy some conditions and λ is large enough, we proved the existence of a positive solution. In particular, we do not assume any symmetric condition, and we do not assume any sign condition on h 1 ( 0 ) and h 2 ( 0 ) .