We study density and extension problems for weighted Sobolev spaces on bounded (ε, δ) domains 𝓓 when a doubling weight w satisfies the weighted Poincaré inequality on cubes near the boundary of 𝓓 and when it is in the Muckenhoupt A p class locally in 𝓓. Moreover, when the weights w i (x) are of the form dist(x, M i )αi , α i∈ ℝ, M i ⊂ 𝓓 that are doubling, we are able to obtain some extension theorems on (ε, ∞) domains.