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 Ap class locally in 𝓓. Moreover, when the weights wi(x) are of the form dist(x, Mi)αi, αi∈ ℝ, Mi⊂ 𝓓 that are doubling, we are able to obtain some extension theorems on (ε, ∞) domains.