Continuity of solutions to the $G$-Laplace equation involving measures - Yan Zhang; Jun Zheng (Electronic Journal of Qualitative Theory of Differential Equations, 2019)
We establish local continuity of solutions to the $G$-Laplace equation involving measures, i.e., \begin{align*} -\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu,\end{align*} where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq Cr^{m}$ for any ball $B_{r}(x_{0})\subset\subset \Omega$ with $r\leq 1$ and $m>n-1-\delta\geq 0$. The function $g$ is supposed to be nonnegative and $C^{1}$-continuous on $[0,+\infty)$, satisfying $g(0)=0$ and \begin{align*}\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0\end{align*} with positive constants $\delta$ and $g_{0}$, which generalizes the structural conditions of Ladyzhenskaya–Ural'tseva for an elliptic operator.