A Harnack inequality in Orlicz–Sobolev spaces
Volume 243 / 2018
Studia Mathematica 243 (2018), 117-137
MSC: Primary 35J20; Secondary 35J60, 35J91.
DOI: 10.4064/sm8764-9-2017
Published online: 8 March 2018
Abstract
A generalized Harnack inequality for the $\phi$-Laplacian $$ -{\rm div}\biggl(\phi(|\nabla u|) \frac{\nabla u}{|\nabla u |}\biggr) = {\mathcal B}(\cdot,u) \ \quad \mbox{in}\ \varOmega $$ is obtained. The domain $\varOmega\subseteq \mathbb R^N$ is bounded and has the segment property. The right-hand side $\mathcal B$ is a Carathéodory function which satisfies mild growth restrictions. The term $\phi$ is an odd and increasing homeomorphism from $\mathbb{R}$ onto $\mathbb{R}$ which is not necessarily differentiable. The lack of smoothness is in striking contrast with the classical case treated by Lieberman.
