Boundary blow-up solutions for a class of elliptic equations on a bounded domain

This paper deals with existence of solutions of the quasilinear elliptic equation Δu = f(x, u, ∇u) in Ω satisfying the boundary blow-up condition u(x)→x→∂Ω∞, where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω. Our main result applies to the existence of blow-up solutions of a class of equations which appear in Stochastic Control Theory, namely Δu = a(x)g(u) + λ∣∇u∣σ + Ψ(x) in Ω, where a(x), g(u), Ψ(x) are suitable functions and λ, σ are nonnegative constants. Our approach employs an approximation procedure combined with both a non-monotone iteration variant of the method of lower and upper solutions and fixed point arguments.