Since the definition of $u\in H^s(\mathbb{R}^n)$ is $\left(1+|\lambda|^2\right)^{s/2}\hat{u}(\lambda)\in L^2(\mathbb{R}^n)$
I find it difficult to give an constructive prove that use mollifier.
let $\rho_\delta$ be a mollifier (for example $\exp(\frac{1}{1-(x/\delta)^2}1_{B(0,\delta)})$ ), then how to say $(u\ 1_{B(0,R)})*\rho_\delta$ converge to $u$ in $H^s$ when $\delta\to 0^+,R\to+\infty$
The main purpose is to prove that $C_0^\infty(\mathbb{R}^n)$ is dense in $E=\{u\in H^s(\mathbb{R}^n):\partial_{x_1}u\in L^2(\mathbb{R}^n)\}$ where $||u||_E^2=||u||_{H^s}^2+||\partial_{x_1}u||_{L^2}^2$. Simply use the fact that $C_0^\infty(\mathbb{R}^n)$ is dense in $H^s(\mathbb{R}^n)$ does not work