Extremal Solutions for a Class of Unilateral Problems

Abstract

We apply a fixed point theorem for increasing operators in ordered Banach spaces to prove the existence of extremal (i.e. maximal or minimal) solutions for the variational inequality $⟨Av,w–v⟩\ge {\int }_{\Omega }f(x,v)(w–v)dx$ where $A$ is the $p$-Laplacian and $f(x,u)=F(x,u,u)$ with $F(x,u,v)$ being a function, non-decreasing in $u$ and non-increasing in $v$.