The absence of global positive solutions of systems of semilinear elliptic inequalities in cones

Let $K$ be a cone in $\mathbb R^N$, $N\geqslant 2$. We establish conditions for the absence of global non-trivial non-negative solutions of semilinear elliptic inequalities and systems of inequalities of the form
$$ -\operatorname{div}(|x|^\alpha Du)\geqslant |x|^\beta u^q, \qquad u\big|_{\partial K}=0. $$
We find the critical exponent $q^*$ that divides the domains of existence of these solutions from those of their absence. We prove that in the limiting case $q=q^*$ there are no solutions. The method is to multiply the system by a special factor and integrate the inequalities thus obtained.