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

Abstract

We introduce a class of weak solutions to the quasilinear equation −Δpu=σ|u|p−2u in an open set Ω⊂Rn with p>1, where Δpu=∇⋅(|∇u|p−2∇u) is the p-Laplacian operator. Our notion of solution is tailored to general distributional coefficients σ which satisfy the inequality −Λ∫Ω|∇h|pdx≤〈|h|p,σ〉≤λ∫Ω|∇h|pdx, for all h∈C0∞(Ω). Here 0<Λ<+∞, and 0<λ<(p−1)2−pif p≥2,or0<λ<1 if 0<p<2. As we shall demonstrate, these conditions on λ are natural for the existence of positive solutions, and cannot be relaxed in general. Furthermore, our class of solutions possesses the optimal local Sobolev regularity available under such a mild restriction on σ.We also study weak solutions of the closely related equation −Δpv=(p−1)|∇v|p+σ, under the same conditions on σ. Our results for this latter equation will allow us to characterize the class of σ satisfying the above inequality for positive λ and Λ, thereby extending earlier results on the form boundedness problem for the Schrödinger operator to p≠2.