Large solution to nonlinear elliptic equation with nonlinear gradient terms

Abstract

The aim of this paper is to study the qualitative behavior of large solutions to the following problem { Δ u ± a ( x ) | ∇ u | q = b ( x ) f ( u ) , x ∈ Ω , u ( x ) = ∞ , x ∈ ∂ Ω . Here a ( x ) ∈ C α ( Ω ) is a positive function with α ∈ ( 0 , 1 ) , b ( x ) ∈ C α ( 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 | ∇ u | q and the weight functions a ( x ) and b ( x ) .