On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces - Cavalheiro Albo Carlos (Annales Mathematicae Silesianae, 2024)
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.