Existence and approximation of solutions for a class of degenerate elliptic equations with Neumann boundary condition

In this work we study the equation $Lu=f$, where $L$ is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set ${\Omega}$. We prove the existence and uniqueness of weak solutions in the weighted Sobolev space ${\mathrm{W}}^{1,2}(\Omega , \omega)$ for the Neumann problem. The main result establishes that a weak solution of degenerate elliptic equations can be approximated by a sequence of solutions for non-degenerate elliptic equation