Keller–Osserman type conditions for some elliptic problems with gradient terms

In this paper we consider the elliptic boundary blow-up problems{Δu±g(|∇u|)=f(u)in Ω,u=∞on ∂Ω, where Ω is a smooth bounded domain and the functions f and g are increasing and continuous. Our main concern will be to prove both existence and nonexistence of nonnegative solutions, depending on new integral conditions of Keller–Osserman type involving f and g. We show in particular that the problem with a minus sign may have solutions inclusive for some functions g with slightly superquadratic growth at infinity that is somehow not expected. We also obtain uniqueness of nonnegative solutions in some cases.