We discuss some regularity results for a class of elliptic equations with coefficients that degenerate or explode on a lower dimensional manifold. The prototype is given by \( -div(|y|^aA(x, y)\nabla u) = |y|^af + div(|y|^aF)\quad\text{in}\;\, B_1\subset\mathbb R^d .\) Here \( z = (x, y) \in\mathbb R^{d−n}\times\mathbb R^{n},\, 2 ≤ n ≤ d\) and \( a \in\mathbb R\). The equation is uniformly elliptic far from a characteristic flat manifold of low dimension \( d - n \in [0, d −-2]\quad\) \(\Sigma_0=\left\{z = (x, y)\in\mathbb R^ d\,:\; |y| = 0\right\}.\) Our approach to Hölder \(C^0,\alpha\) and Schauder \(C^{1,\alpha}\) estimates involves approximation via perforated domains, blow-up analysis and Liouville- type Theorems. This is a joint project with Gabriele Cora and Gabriele Fioravanti.