On the Sub and Supersolution Method for Nonlinear Elliptic Equations with a Convective Term, in Orlicz Spaces

Abstract

In this note we provide an overview of some existence (with sign information) and regularity results for differential equations, in which the method of sub and supersolutions plays an important role. We list some classical results and then we focus on the Dirichlet problem, for problems driven by a general differential operator, depending on (x,u,∇u), and with a convective term f. Our framework is that of Orlicz–Sobolev spaces. We also present several examples.

Keywords

• nonlinear elliptic equations
• Orlicz–Sobolev spaces
• gradient dependence
• subsolution and supersolution