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We introduce a class of weak solutions to the quasilinear equation $-\Delta_pu = \sigma |u|^{p-2}u$ 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 istailored to general distributional coefficients $\sigma$ satisfying a certainweighted Sobolev-Poincare inequality. We also study weak solutions of theclosely related equation $-\Delta_p v = (p-1)|\nabla v|^p + \sigma$, under thesame conditions on $\sigma$. Our results for this latter equation will allow usto characterize the class of distributions $\sigma$ which satisfy theSobolev-Poincare inequality, thereby extending earlier results on the formboundedness problem for the Schr\"odinger operator to $p\neq 2$.

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