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: $-Δ_{p,a} u:= -\mathrm{div}(a(x)|\na u|^{p-2}\na u)\ge b(x)Φ(u)$, where $u$ is defined in a domain $Ω$. Using Caccioppoli--type estimates, we obtain several variants of Hardy--type inequalities in weighted Sobolev spaces.

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