I define $\gamma(x)=e^{-x^2}$. I am interested in the $W_0^{1,2}(\mathbb{R},\gamma)$ which I define as the completion of compactly supported functions, for the $\|.\|_ {W_{0,\gamma}^{1,2}}$ defined by $$\|u\|_ {W_{0,\gamma}^{1,2}}:=\|u\| _{L^2(\gamma)}+\|u'\| _{L^2(\gamma)}$$
My question is: Is it true that the functions in $W_0^{1,2}(\mathbb{R},\gamma)$ contains only functions that vanish at infinity ? It is true when the Sobolev space is not weighted but I can't find a reference which states it properly. I've only found articles where people say that set contains only functions which are vanishing at infinity See Part4 of https://link.springer.com/content/pdf/10.1007/s10013-020-00431-1.pdf