I am having trouble solving a linear degenerate elliptic equation. The problem is as follows. Let $U\subset \mathbb{R}^n$ be a bounded open set and $\omega:U\to\mathbb{R}$ is a $C^\infty$ function such that $\omega(x) = Cd(x,\partial U)$, where $C>0$. Define the Weighted Sobolev space $H^{s,\sigma}(U)$ as the space of distributions $f$ s.t the following norm is finite,

$$\lVert f \rVert_{H^{k,\sigma}(U)}^2 = \sum_{i=1}^k \|\omega^\sigma\partial_i f \|_{L^2(U)}^2.$$

Given $f\in H^{0,\sigma}$, I want to solve the following linear degenerate elliptic equation for $u\in H^{2,\sigma+1},

$$-\delta^{ij}[\omega\partial_i\partial_j u + (2\sigma+1)\partial_i \omega\partial_j u] + \omega u = f.$$

Can anyone provide any reference that would help to solve this problem?

I am having trouble solving a linear degenerate elliptic equation. The problem is as follows. Let $U\subset \mathbb{R}^n$ be a bounded open set and $\omega:U\to\mathbb{R}$ is a $C^\infty$ function such that $\omega(x) = Cd(x,\partial U)$, where $C>0$. Define the Weighted Sobolev space $H^{s,\sigma}(U)$ as the space of distributions $f$ s.t the following norm is finite,

$$\lVert f \rVert_{H^{k,\sigma}(U)}^2 = \sum_{i=1}^k \|\omega^\sigma\partial_i f \|_{L^2(U)}^2.$$

Given $f\in H^{0,\sigma}$, I want to solve the following linear degenerate elliptic equation for $u\in H^{2,\sigma+1}$,

$$-\delta^{ij}[\omega\partial_i\partial_j u + (2\sigma+1)\partial_i \omega\partial_j u] + \omega u = f.$$

Can anyone provide any reference that would help to solve this problem?