Let
\begin{equation*} \eta(x)=C_n \exp\Bigl(-\frac{1}{1-\lvert x\rvert^2}\Bigr) 𝟙_{B_1(0)}, \end{equation*}where \(C_n\) ensures \(\int_{\mathbb{R}^n} \eta =1\). We call \(\eta\) standard mollifier. The standard mollifier is compactly supported \(B_1(0)\) and smooth.
For \(\varepsilon>0\) we define
\begin{equation*} \eta_{\varepsilon}(x)=\frac{1}{\varepsilon^n}\eta\Bigl(\frac{x}{\varepsilon}\Bigr). \end{equation*}It is compactly supported on \(B_{\varepsilon}(0)\) and it is smooth.