Regularity for infinitely degenerate inhomogenous elliptic equations, Part I: The Moser Method

Abstract

We show that if $R^n$ is equipped with a 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 equations of the form $-\mathrm{div}A\left(x,u\right)\nabla u={\varphi }_0-{\mathrm{div}}_A{\overrightarrow{\varphi }}_1$ are locally bounded. This is obtained by the implementation of a Moser iteration method, what constitutes the first instance of such technique applied to infinite degenerate equations. The results presented here partially extend previously known estimates for solutions of similar equations in which the right hand side does not have a drift term. We also obtain bounds for small negative powers of nonnegative solutions, which will be applied in a subsequent paper to prove continuity of solutions. We also provide examples of geometries in which our abstract theorem is applicable. We consider the family of functions $f_{k,\sigma }\left(x\right)={\left(x\right)}^{{\left({ln}^{\left(k\right)}\frac{1}{\left(x\right)}\right)}^{\sigma }},k\in N,\sigma >0,-\infty <x<\infty ,$ infinitely degenerate at the origin, and show that all weak solutions to $-\mathrm{div}A\left(x,y,u\right)\nabla u=\varphi \left(x,y\right)-{\mathrm{div}}_A{\overrightarrow{\varphi }}_1\left(x,y\right),A\left(x,y,z\right)\sim \left(\begin{array}{cc}1& 0\\ 0& f_{k,\sigma }{\left(x\right)}^2\end{array}\right),$ with rough data $A,{\varphi }_0,{\overrightarrow{\varphi }}_1$, are locally bounded when $k=1$ and $0<\sigma <1$.