On equivalent conditions for the validity of Poincaré inequality on weighted Sobolev space with applications to the solvability of degenerated PDEs involving p-Laplacian

Let Ω⊆RnΩ⊆Rn be a given subset, p>1p>1 and Wρ1,p(Ω)={f∈Lloc1(Ω):f,∂f∂x1,…,∂f∂xn∈Lρp(Ω)}, be the weighted Sobolev space subordinated to the weight function ρ  . We derive a chain of equivalent conditions for the validity of Poincaré inequality (P):∫Ω|f(x)|pρ(x)dx≤C∫Ω|∇f(x)|pρ(x)dx. The conditions include: isoperimetric inequalities, representation of functionals, as well as solvability of degenerated elliptic problem (S):div(ρ(x)|∇u|p−2∇u)=x⁎ where x⁎x⁎ belongs to the dual space to the completion of C0∞(Ω) in Wρ1,p(Ω). Furthermore, we adopt recent technique due to Skrzypczak to construct one parameter families of weights ρβρβ such that the problem (S) is solvable with ρ=ρβρ=ρβ.