On blow-up conditions for solutions of higher order differential inequalities

For differential inequalities of the form $$\sum _{\mathrm{\mid }\alpha \mathrm{\mid }=m}(-1{)}^m{\mathrm{\partial }}^{\alpha }{a}_{\alpha }(x,u)\ge b(x)\mathrm{\mid }u{\mathrm{\mid }}^{\lambda }in{\mathbb{R}}^n,n\ge 1,$$ where $a_{\alpha }$ and $b$ are some functions, we obtain conditions guaranteeing that any solution is identically equal to zero. We construct examples which show that the obtained conditions are sharp.