Barrier solutions of elliptic differential equations in Musielak-Orlicz-Sobolev spaces (Q2236569)

Summary: In this paper, we study the solution set of the following Dirichlet boundary equation: \(-\operatorname{div}(a_1 (x, u, Du))+ a_0(x, u)=f(x, u, Du)\) in Musielak-Orlicz-Sobolev spaces, where \(a_1:\Omega\times\mathbb{R}\times \mathbb{R}^N\longrightarrow \mathbb{R}^N\), \(a_0:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}\), and \(f:\Omega\times\mathbb{R}\times \mathbb{R}^N\longrightarrow\mathbb{R}\) are all Carathéodory 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 ``\(\leq\)'', which are called barrier solutions.