On a Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Spaces

Abstract

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

$ \begin{aligned}& -\sum_{j=1}^{n} D_{j}\left[\omega(x) \mathcal{B}_{j}(x, u, \nabla u)\right]-\sum_{i, j=1}^{n} D_{j}\left(a_{i j}(x) D_{i} u(x)\right)+b(x, u, \nabla u) \omega(x)+g(x) u(x) \\= & f_{0}(x)-\sum_{j=1}^{n} D_{j} f_{j}(x), \quad \text { in } \Omega\end{aligned}$

in the setting of the weighted Sobolev spaces $ W_{0}^{1, p}(\Omega, \omega).$

Keywords

Degenerate nonlinear elliptic equations / weighted Sobolev spaces / ApAp-weights