In this paper, our main purpose is to establish the existence of positive solution
of the following system
v= H(x,u,v), x∈Ω,
where Ω = B(0,r) ⊂ R
N or Ω = B(0,r2)\B(0,r1) ⊂ R
N, 0 < r, 0 < r1 < r2 are constants.
[g(x)a(u)+ f(v)], H(x,u,v)=θ
[g1(x)b(v)+h(u)], λ,θ>0 are parameters,
p(x), q(x) are radial symmetric functions, −∆p(x)=−div(|∇u|
p(x)−2∇u) is called
p(x)-Laplacian. We give the existence results and consider the asymptotic behavior of
the solutions. In particular, we do not assume any symmetric condition, and we do
not assume any sign condition on F(x,0,0) and H(x,0,0) either.