The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

We prove the existence of extremal solutions of the following quasilinear elliptic problem - Sigma(N)(i=1) (partial derivative/partial derivative(i))a(i)u(x,u)(x), Du(x)) + g(x, u(x), Du(x)) = 0 under Dirichlet boundary condition in Orlicz-Sobolev spaces W0LM(Omega) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term g : Omega x R x R-N -> R is a Caratheodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.