The sub-supersolution method and extremal solutions of quasilinear elliptic equations in Orlicz-Sobolev spaces
Summary: We prove the existence of extremal solutions of the following quasilinear elliptic problem \(- \sum_{i = 1}^N (\partial / \partial x_i) a_i(x, u(x), D u(x)) + g(x, u(x), D u(x)) = 0\) under Dirichlet boundary condition in Orlicz-Sobolev spaces \(W_0^1 L_M(\Omega)\) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term \(g : \Omega \times \mathbb{R} \times \mathbb{R}^N \rightarrow \mathbb{R}\) is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.
scientific article; zbMATH DE number 6933358
scientific article