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

Abstract

We introduce a class of weak solutions to the quasilinear equation $-{\Delta }_{p}u=\sigma |u{|}^{p-2}u$ in an open set $\Omega \subset {\mathbf{R}}^n$. Here $p>1$, and ${\Delta }_{p}u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $\sigma$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-{\Delta }_{p}v=(p-1)|\nabla v{|}^p+\sigma$, under the same conditions on $\sigma$. Our results for this latter equation will allow us to characterize the class of distributions $\sigma$ which satisfy the Sobolev-Poincare inequality, thereby extending earlier results on the form boundedness problem for the Schrödinger operator to $p\ne 2$.

36 pages