Some Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces

In this note we deal with the problem of the density of smooth functions in weighted Sobolev spaces extending research of O. V. Besov ["On the denseness of the compactly supported functions in a weighted Sobolev space", Proc. Steklov Inst. Mat. 161 (1984) 33–52]. Using the common terminology we show in particular that if $n = 2$, then for every $p > 1$ the equality between $W^{1,p}(\Omega, \lambda)$ and $H^{1,p}(\Omega, \lambda)$ does not hold any more for a suitable weight $\lambda$.