Large solution to nonlinear elliptic equation with nonlinear gradient terms

The aim of this paper is to study the qualitative behavior of large solutions to the following problem{Δu±a(x)|∇u|q=b(x)f(u),x∈Ω,u(x)=∞,x∈∂Ω. Here a(x)∈Cα(Ω)a(x)∈Cα(Ω) is a positive function with α∈(0,1)α∈(0,1), b(x)∈Cα(Ω)b(x)∈Cα(Ω) is a non-negative function and may be singular near the boundary or vanish on the boundary, and the nonlinear term f is a Γ-varying function, whose variation at infinity is not regular. We focus our investigation on the existence and asymptotic behavior close to the boundary ∂Ω   of large solutions by Karamata regular variation theory and the method of upper and lower solution. The main results of this paper emphasize the central role played by the gradient term q|∇u||∇u|q and the weight functions a(x)a(x) and b(x)b(x).