The Dirichlet Problem of Nonhomogeneous Degenerate Nonlinear Elliptic Equations

Abstract

In this paper we are interested in the existence and uniqueness of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations

\{\begin{array}{c}-\text{div}[\mathcal{A}(x,\mathrm{\nabla }u)v(x)]+\mathcal{B}(x,u)\omega (x)={\rho }_0-\sum _{j=1}^n{D}_j{\rho }_j,\hfill \\ u-\psi \in W_0^{1,p}(\mathrm{\Omega },\omega ,v),\hfill \end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left\{{\matrix{{- {\text{div}}[{\cal{A}}(x,\nabla u)v(x)] + {\cal{B}}(x,u)\omega (x) = {\rho _0} - \sum\limits_{j = 1}^n {{D_j}{\rho _j},}} \hfill \cr {u - \psi \in W_0^{1,p}(\Omega,\omega,v),} \hfill \cr}} \right.$$\end{document}

in the setting of the weighted Sobolev spaces.

Keywords

Degenerate nonlinear elliptic equations / weighted Sobolev spaces / 35J70 / 35J60 / 35J30

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