L∞-Solutions for Some Nonlinear Degenerate Elliptic Equations

Abstract

We are interested in the existence of solutions for Dirichlet problem associated to the degenerate quasilinear elliptic equations $-{\sum }_{j=1}^n{D}_j\left[{\omega }_2\right(x\left){\mathcal{A}}_j\right(x,u,\nabla u\left)\right]+{\omega }_1\left(x\right)g(x,u(x\left)\right)+H(x,u,\nabla u){\omega }_2\left(x\right)=$$f\left(x\right),\text{on}\Omega$ in the setting of the weighted Sobolev spaces ${\text{W}}_0^{1,p}(\Omega ,{\omega }_1,{\omega }_2)$.