Hardy–Poincaré type inequalities derived from p-harmonic problems

Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

“…As it is shown in [68], the certain substitution in the derived Caccioppolitype inequality for solutions, implies the family of Hardy-type inequalities having the form…”

“…The involved measures μ 1,β (dx), μ 2,β (dx) depend on a certain parameter β and on u-a nonnegative weak solution to (1.7). It leads among other results to classical Hardy and Hardy-Poincaré inequalities with optimal constants (see [68,69], respectively). We retrieve this results as the special case here and therefore we confirm all the examples from [68,69].…”

“…Our goal now is to extend techniques from [68] to the more general situation when we deal with (1.2) instead of (1.7). We supply new weighted power-logarithmic Hardy-Sobolev inequalities of the form…”

“…Futhermore, in papers of D'Ambrosio [23][24][25] the author derives inequality related to (1.1) involvingĀ(λ) = FĀ(λ) = λ p as the consequence of inequality −Δ p (u α ) ≥ 0 with certain constant α. Our cosiderations are based on the methods from [47] developed further in [68]. The idea is as follows.…”

“…The proof is based on the idea of the proof of Theorem 3.1 from [68] inspired by the proof of Proposition 3.1 from [47].…”