On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces

Abstract

In this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations − div𝒜x,u,Δu ω1+ℬx,u,∇uν1+ℋx,u,∇uν2+up−2 u ω2−∑i,j=1nDjaijxDiux=f0x−∑j=1nDjfjx in Ω,ux=0 on ∂Ω,\begin{array}{*{20}{c}}{ - \;{\rm{div}}\left[ {\mathcal{A}\left( {x,u,\Delta u} \right)\;{\omega _1} + {\mathcal{B}}\left( {x,u,\nabla u} \right){\nu _1}} \right] + \mathcal{H}\left( {x,u,\nabla u} \right){\nu _2} + {{\left| u \right|}^{p - 2}}u\;{\omega _2}}\\{ - \sum\limits_{i,j = 1}^n {{D_j}\left( {{a_{ij}}\left( x \right){D_i}u\left( x \right)} \right)} = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)} \;\;\;\;\;{\rm{in}}\;\;\Omega ,}\\{u\left( x \right) = 0\;\;\;\;\;{\rm{on}}\;\partial \Omega ,}\end{array} in the setting of the weighted Sobolev spaces.

Keywords

• 35j70
• 35j60
• 35j30
• degenerate nonlinear elliptic equations
• weighted sobolev spaces