Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

In this paper, we study the solution set of the following Dirichlet boundary equation: -div (a(1)(x, u, Du)) + a(0)(x, u) = f (x, u, Du) in Musielak-Orlicz-Sobolev spaces, where a(1) : Omega x R x R-N -> R-N, a(0) : Omega x R -> R, and f : Omega x R x R-N -> R are all Caratheodory functions. Both a(1) and f depend on the solution u and its gradient Du. By using a linear functional analysis method, we provide sufficient conditions which ensure that the solution set of the equation is nonempty, and it possesses a greatest element and a smallest element with respect to the ordering "<=," which are called barrier solutions.