Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces. (7th October 2021) - Dong, Ge; Fang, Xiaochun
Abstract : In this paper, we study the solution set of the following Dirichlet boundary equation: − div a 1 x, u, D u + a 0 x, u = f x, u, D u in Musielak-Orlicz-Sobolev spaces, where a 1 : Ω × ℝ × ℝ N ⟶ ℝ N, a 0 : Ω × ℝ ⟶ ℝ, and f : Ω × ℝ × ℝ N ⟶ ℝ are all Carathéodory functions. Both a 1 and f depend on the solution u and its gradient D u . 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.