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, $$\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^p(v,\mathrm{\Omega })}\le S(p,1)\mathrm{\parallel }\sqrt{Q}\mathrm{\nabla }\phi {\mathrm{\parallel }}_{L^p(\mathrm{\Omega })},$$ that holds for smooth functions of compact support and the existence of a degenerate weak solution $(u,\mathrm{\nabla }u)\in Q{H}_0^{1,p}(v,\mathrm{\Omega })$ to a Dirichlet problem for the $p$-Laplacian with a zero order term: \begin{equation*} -v^{-1}$\text{Div}$(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)+F|u|^{p-2}u = |f|^{p-2}f - v^{-1}$\text{Div}$(v|g|^{p-2}g$\mathbf{t}$),; x \in Ω, \quad $\text{and}$ \quad u =0, ; x \in \partial Ω, \end{equation*} More precisely, we use the Sobolev inequality to prove the existence of a degenerate weak solution to this equation and then use the existence of such a solution to produce a Sobolev inequality. Moreover, we show that solutions are unique.