Harnack's inequality for quasilinear elliptic equations with singular absorption term
In this interesting paper, the author is concerned with divergence type quasilinear elliptic equations with measurable coefficients and singular lower order term. More precisely, the author studies Harnack inequality of nonnegative solutions to the equations \[ -\operatorname{div}A(x, \nabla u) + a_0(x, u) = b(x, \nabla u),\quad x \in \Omega, \] where $\Omega$ is a domain in $\mathbb{R}^n$, $n \geq 2$, $A$ is an operator with a $p$-Laplacean structure, $a_0$ is a lower term and $b$ is an absorption term with singular coefficients. To treat the quasilinear case of $p$-Laplacian with a lower order term, Biroli introduced the notion of the nonlinear Kato class. His method was extended to more general cases by using the Kilpelainen-Maly technique. In this paper the author extends these results to the case of elliptic equations with absorption term. The difficulty is that neither De Giorgi nor Moser iteration techniques work in this situation. In this paper, by using the Kilpelainen-Maly technique, the author establishes the Harnack inequality for the $p$-Laplace equation perturbed by absorbtion term with singular coefficient.
divergence type quasilinear elliptic equations
scientific article; zbMATH DE number 7049464
scientific article