On the weak Harnack inequality for quasilinear elliptic equations

L. V. Davydova

Sbornik. Mathematics, Tome 53 (1986) no. 2, pp. 335-349

Résumé

A generalization of Harnack's inequality is given for solutions of the differential inequality \begin{equation} |Lu|\leqslant K_1|\nabla u|^{1+\alpha}+K_2, \end{equation} in which $L$ is a uniformly elliptic operator with measurable and bounded coefficients, $K_1$ and $K_2$ are fixed positive constants, and $\alpha$, $0\alpha1$, is some number. It is shown that there exist $\alpha_0$, $0\alpha_01$, depending on the ellipticity constant and the dimension of the space, and $M_0>1$, depending on the ellipticity constant, the dimension of the space and the numbers $K_1$, $K_2$ and $\alpha$, such that for solutions $u$ of inequality (1) with $\alpha\alpha_0$ which are positive in the ball of radius $R$ with center at the origin, and such that $u(0)=M>M_0$, Harnack's inequality holds if $R$ is commensurate with $M^{-\alpha/(1-\alpha_0)}$ with the constant in Harnack's inequality depending only on the dimension of the space and the ellipticity constant. Bibliography: 9 titles.