Some estimates of solutions of degenerate $(k,0)$-elliptic equations

L. P. Kuptsov

Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 625-634

Résumé

A class of nonlinear second-order equations of divergent form is distinguished, whose solutions have properties recalling the properties of solutions of ordinary elliptic equations. In the linear case these are equations of the form $$ \sum_{j=1}^k\lambda_j(x)A_j^2u+\sum_{j=1}^k\mu_j(x)A_ju+c(x)u+f(x)=0 $$ where the $A_j=\sum_{\alpha=1}^na_j^\alpha(x)\frac\partial{\partial x^\alpha}$ ($1\le j\le k$) are linearly independent first-order differential operators whose Lie algebra is of rank $n$, $2\le k\le n$, $\lambda_j(x)\ge0$ are functions which can become zero or increase in a definite way. Harnack's inequality is proved for nonnegative solutions of these equations.