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

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.

Existence of Solutions for Inclusion Problems in Musielak-Orlicz-Sobolev Space Setting

“…In recent years, there are many results of differential equations in Musielak-Orlicz-Sobolev spaces. For example, Fan [17] and we [18] proved the existence of weak solutions of a class of differential equations of divergence form by using a subsupersolution method in reflexive Musielak-Orlicz-Sobolev spaces and nonreflexive Musielak-Orlicz-Sobolev spaces, respectively; Li et al [19] proved the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces; we [20] proved the existence of barrier solutions of elliptic differential equations in Musielak-Orlicz-Sobolev spaces; and Baasandorj et al [21] established optimal regularity estimates for the gradient of solutions to nonuniformly elliptic equations of the Orlicz double phase with variable exponent types. Musielak-Orlicz functions (described in Section 2) have many applications such as non-Newtonian fluids (see, e.g., [22]), thermistor problem (see, e.g., [23]), and image restoration (see, e.g., [24]).…”