The Moser method and boundedness of solutions to infinitely degenerate elliptic equations

Abstract

We show that if $\mathbb{R}^{n}$ is equipped with certain non-doubling metric and an Orlicz-Sobolev inequality holds for a special family of Young functions $\Phi $, then weak solutions to quasilinear infinitely degenerate elliptic divergence equations of the form $$\mathrm{div}\mathcal{A}\left( x,u\right) \nabla u=\phi _{0}-\mathrm{div}_{A} \vec{\phi}_{1}$$ are locally bounded. Furthermore, we establish a maximum principle for solutions whenever a global Orlicz-Soblev estimate is available. We obtain these results via the implementation of a Moser iteration method, what constitutes the first instance of such technique applied to infinite degenerate equations. These results partially extend previously known estimates for solutions of these equations but for which the right hand side did not have a drift term. We also obtain bounds for small negative powers of nonnegative solutions; these will be applied to obtain continuity of solutions in a subsequent paper.