We introduce the following functional to study Yamabe problem with boundary.

$$ Q_g(\varphi)=\frac{\int_M\left(|\nabla \varphi|_g^2+\frac{n-2}{4(n-1)} R_g \varphi^2\right) d v+\frac{n-2}{2} \int_{\partial M} h_g \varphi^2 d \sigma}{\left(\int_M|\varphi|^{2 n /(n-2)} d v\right)^{(n-2) / 2}} $$

$$ Q(M)=\inf \left\{Q_g(\varphi): \varphi \in C^1(\bar{M}), \varphi \neq 0\right\} $$

In Escobar's The Yamabe Problem On Manifolds With Boundary,he says if $Q(M)<Q(S^{+}_{n})$,the functional has a critical point which is the solution of Yamabe Problem.

He also gives the formula of $Q(S_{n}^{+})$

$$ Q\left(S_{+}^n\right)=\frac{\int_{\mathbb{R}_{+}^n}\left|\nabla u_{\varepsilon}\right|^2}{\left(\int_{\mathbb{R}_{+}^n} u_{\varepsilon}^{2 n /(n-2)}\right)^{(n-2) / n}}=n(n-2)\left(\int_{\mathbb{R}_{+}^n} u_{\varepsilon}^{\frac{2 n}{n-2}}\right)^{2 / n}, $$ $$ u_{\varepsilon}\left(x, x_n\right)=\left(\frac{\varepsilon}{\varepsilon^2+|x|^2+x_n^2}\right)^{(n-2) / 2} $$

I wonder how this is derived.

When studying the Yamabe problem on compact manifolds,we use conformal change to calculate $Q(S_{n})$.Since the functional is conformally invariant, we use conformal change to transfer the problem to $R^{n}$,which becomes the problem of best Sobolev constant.

How can we do this when studying the problem on manifolds with boundary?Any help will be thanked.