Sobolev Inequalities and the Existence of Solutions to Degenerate $p$-Poisson Equations

In this paper we study an equivalence between the existence of a Sobolev inequality without gain, \[\|φ\|_{L^p(v,Ω)} \leq S(p,1) \| \sqrt{Q}\nabla φ\|_{L^p(Ω)},\] that holds for smooth functions of compact support and the existence of a degenerate weak solution $(u,\nabla u)\in QH^{1,p}_0(v,Ω)$ to a