Existence and uniqueness of solutions of degenerate elliptic equations with lower order terms

Abstract

We prove the existence and uniqueness of solutions to a Dirichlet problem \[ \begin{cases} Lu = f + v^{-1}\text{Div}(v{\bf e} h), & x \in \Omega; u = 0, & x \in \partial \Omega, \end{cases}\] where $L$ is a degenerate, linear, second order elliptic operator with lower order terms. We assume very weak hypotheses, in terms of the coefficients of the equation, and we also assume the existence of degenerate Sobolev and Poincar\'e inequalities. One notable feature of our result is that we show that we can assume significantly weaker versions of the Sobolev inequality if we in turn assume stronger integrability conditions on the coefficients. Our theorems generalize a number of results in the literature on degenerate elliptic equations.