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

We introduce a class of weak solutions to the quasilinear equation $-\Delta_p u = \sigma |u|^{p-2}u$inanopenset$\Omega\subset$\mathbf{R}$^n$\mathrm{.}Here$p>1$, and ${\mathrm{\Delta }}_{p}u$ 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 $-{\mathrm{\Delta }}_{p}v=(p-1)\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^p+\sigma$, 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 $p\mathrm{\ne }2$.