Quasilinear elliptic equations and weighted Sobolev–Poincaré inequalities with distributional weights

We introduce a class of weak solutions to the quasilinear equation −Δpu=σ|u|p−2u−Δpu=σ|u|p−2u in an open set Ω⊂Rn with p>1p>1, where Δpu=∇⋅(|∇u|p−2∇u)Δpu=∇⋅(|∇u|p−2∇u) is the pp-Laplacian operator. Our notion of solution is tailored to general distributional coefficients σσ which satisfy the inequality −Λ∫Ω|∇h|pdx≤〈|h|p,σ〉≤λ∫Ω|∇h|pdx,−Λ∫Ω|∇h|pdx≤〈|h|p,σ〉≤λ∫Ω|∇h|pdx, for all h∈C0∞(Ω). Here 0<Λ<+∞0<Λ<+∞, and 0<λ<(p−1)2−pif p≥2,or0<λ<1 if 0