Existence of boundary blow-up solutions for a class of quasilinear elliptic systems with critical case

We consider the quasilinear elliptic systemdiv(|∇u|p-2∇u)=um1vn1,div(|∇v|q-2∇v)=um2vn2inΩ,where m1>p-1,n2>q-1,m2,n1>0m1>p-1,n2>q-1,m2,n1>0, and Ω⊂RNΩ⊂RN is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: u=λu=λ, v=μoru=v=+∞ or u=+∞,v=μu=+∞,v=μ on ∂Ω∂Ω, where λ,μ>0λ,μ>0. Under several hypotheses on the parameters m1,n1,m2,n2m1,n1,m2,n2 which is a critical case, we show that the existence of positive solutions.