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 {−Δp(x)u=λp(x)(uα(x)vγ(x)+h1(x)) in Ω,−Δq(x)v=λq(x)(uδ(x)vβ(x)+h2(x)) in Ω,u=v=0 on ∂Ω, where Ω⊂RNΩ⊂RN is a bounded domain with C2C2 boundary ∂Ω∂Ω, 1