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

Abstract

In this paper we study an equivalence between the existence of a Sobolev inequality without gain, \[\|\varphi\|_{L^p(v,\Omega)} \leq S(p,1) \| \sqrt{Q}\nabla \varphi\|_{L^p(\Omega)},\] that holds for smooth functions of compact support and the existence of a degenerate weak solution $(u,\nabla u)\in QH^{1,p}_0(v,\Omega)$ 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 \Omega, \quad \text{and} \quad u =0, \; x \in \partial \Omega, \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.