For every compact subset $K\subset \mathbb R^n$ and every $\epsilon>0$ $\exists$ a test function $\psi\in C_c^{\infty}(\mathbb R^n)$ such that
(a) $0\le\psi(x)\le 1\forall x\in \mathbb R^n$
(b) $\psi(x)=1\forall x\in K$
(c) $\psi(x)=0\forall x\in \mathbb R^n/K_{\epsilon}$ for some $K_{\epsilon}$
This lemma is given here on page 7,but the proof is compactly written and I'm trying to detailing it ...
Since $\varphi$ is the scalar multiple of $u(x)$.So, $\varphi$ can be written as $\varphi=K.u(x)$,where $K\in \mathbb R$
Now I've to show that $\int_{\mathbb R^N}\varphi =1$.For this I've made an attempt but I got stuck and unable to proceed further.Please give it a glance
$K.\int\int\int...\int e^{\frac{-1}{1-x_1^2-x_2^2-x_3^2-...-x_N^2}}dx_1dx_2dx_3...dx_N$
Let $x_1^2=u_1\implies x_1=u_1^{1/2}\implies dx_1=\frac{1}{2}u_1^{-1/2}du_1$. Similarly, $dx_2=\frac{1}{2}u_2^{-1/2}du_2$,$dx_3=\frac{1}{2}u_3^{-1/2}du_3$,$dx_4=\frac{1}{2}u_4^{-1/2}du_4$,...,$dx_N=\frac{1}{2}u_N^{-1/2}du_N$.So,it become
$K.\int\int\int...\int e^{\frac{-1}{1-u_1-u_2-u_3-...-u_N}}\frac{1}{2^N}u_1^{\frac{-1}{2}}u_2^{\frac{-1}{2}}u_3^{\frac{-1}{2}}...u_N^{\frac{-1}{2}}du_1du_2du_3...du_N$
$$= K.\frac{1}{2^N}\int\int\int...\int e^{\frac{-1}{1-u_1-u_2-u_3-...-u_N}}u_1^{\frac{1}{2}-1}u_2^{\frac{1}{2}-1}u_3^{\frac{1}{2}-1}...u_N^{\frac{1}{2}-1}du_1du_2du_3...du_N$$ By Liouville's extension of Dirichlet's theorem for multiple integrals,we have $$=K.\frac{1}{2^N}\frac{{\Gamma{(\frac{1}{2})}}{\Gamma{(\frac{1}{2})}}{\Gamma{(\frac{1}{2})}}{\Gamma{(\frac{1}{2})}}...{\Gamma{(\frac{1}{2})}}}{{\Gamma({(\frac{1}{2})+(\frac{1}{2})...+(\frac{1}{2})})}}\int_0^1 e^{\frac{1}{1-u}}u^{1/2+1/2+1/2+...1/2-1}du$$
$=K.\frac{1}{2^N}\frac{(\Gamma(1/2))^N}{\Gamma(N/2)}\int_0^1 e^{\frac{-1}{1-u}}u^{N/2-1}du$
$=K.\frac{1}{2^N}\frac{(\sqrt \pi)^N}{\Gamma(N/2)}\int_0^1 e^{\frac{-1}{1-u}}u^{N/2-1}du$
From here unable to proceed further