Caccioppoli--type estimates and Hardy--type inequalities derived from degenerated $p$--harmonic problems

We obtain Caccioppoli–type estimates for nontrivial and nonnegative solutions to the anticoercive partial differential inequalities of elliptic type involving degenerated $p$–Laplacian: $-\Delta_{p,a} u:= -$\mathrm{d}\mathrm{i}\mathrm{v}$(a(x)|\na u|^{p-2}\na u)\ge b(x)\Phi (u)$,where$u$ is defined in a domain $\mathrm{\Omega }$. Using Caccioppoli–type estimates, we obtain several variants of Hardy–type inequalities in weighted Sobolev spaces.