Quasilinear elliptic equations and weighted Sobolev-Poincar\'{e} inequalities with distributional weights

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 Schr\"odinger operator to $p\neq 2$.