Harnack Inequalities and Continuity of Solutions under Exponential Logarithmic Orlicz-Sobolev Inequalities

Lyudmila Korobenko, Cristian Rios, Eric Sawyer, Ruipeng Shen

Published 2023-03-06, updated 2023-05-30Version 2

In dimensions $n\geq 2$ we implement the Moser iteration in $\mathbb{R}^{n}$ endowed with a certain metric $d$ which is topologically equivalent to the Euclidean metric, but for which Lebesgue's measure is not doubling. In this metric space we prove Harnack's inequality for weak solutions to infinitely degenerate quasilinear elliptic equations $-\mathrm{div}\mathcal{A}\left( x,u\right) \nabla u=\phi _{0}-\mathrm{div}_{A}\vec{\phi}_{1}$.