Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions

We develop a sub- and supersolution method for prescribed mean curvature equations with Dirichlet boundary conditions. Our result is based on the work of Noussair, Swanson and Yang (1993) and essentially improve the method proposed by them. We remove some unnecessary assumptions from their main results, thus make the method much easier to use and broaden its range of applications. Further, we apply the improved sub- and supersolution method to several concrete examples such as a MEMS model, a Liouville–Gelfand type problem, and a sublinear problem. These applications give some new existence results and reflect the advantages of the improved method.