On the properties of solutions of nonlinear elliptic inequalities containing terms with lower order derivatives (Q416925)

This paper deals with the nonnegative solutions of the inequality \[ \text{div\,}a(x, Du)+ b(x)|Du|^{p-1}\geq q(x) g(u)\quad\text{in }\Omega,\tag{*} \] \(p> 1\), \(u|_{\partial\Omega}= 0\), \(\Omega\) being unbounded domain in \(\mathbb{R}^n\), \(n\geq 2\) and the form \(a(x,\xi)\) being locally Carathéodory and elliptic in \(\xi\). Under appropriate conditions on the coefficients \(b\), \(q\) it is shown in Theorem 1 that any nonnegative solution of (*) is zero. Assuming that one of that conditions is violated the author proved an estimate from below for any nontrivial nonnegative solution of (*). The paper is illustrated by 7 examples and it is mentioned that all estimates given in Examples 5--7 are sharp. It can be considered as a continuation and generalization of previous results of the same author.