Functional of fully nonlinear equations

Question

Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is to prove the existence of a conformal metric $g=g_v=v^{\frac{4}{n-2}} g_0$ that solves the equation $$ \sigma_k\left(\lambda\left(A_g\right)\right)=1 \quad \text { on } \mathcal{M} $$ where $1 \leq k \leq n$ is an integer, and $\lambda=\left(\lambda_1, \ldots, \lambda_n\right)$ are the eigenvalues of $A_g$ with respect to the metric $g$. As usual, we denote by $$ A_g=\frac{1}{n-2}\left(\operatorname{Ric}_g-\frac{R_g}{2(n-1)} g\right) $$ the Schouten tensor, and by $$ \sigma_k(\lambda)=\sum_{i_1<\cdots<i_k} \lambda_{i_1} \cdots \lambda_{i_k} $$the Schouten tensor is given by $A_g=\frac{2}{(n-2) v} V$, and $v$ satisfies the equation $$ L[v]:=v^{(1-k) \frac{n+2}{n-2}} \sigma_k(\lambda(V))=v^{\frac{n+2}{n-2}}, $$ where $$ V=-\nabla^2 v+\frac{n}{n-2} \frac{\nabla v \otimes \nabla v}{v}-\frac{1}{n-2} \frac{|\nabla v|^2}{v} g_0+\frac{n-2}{2} v A_{g_0} $$My question is how to prove that when $k<\frac{n}{2}$, equation $Lv=v^{\frac{n+2}{n-2}}$ is the Euler equation of the follwing functional $$ \begin{aligned} J_p(v) & =J_p(v ; \mathcal{M}) \\ & =\frac{n-2}{2 n-4 k} \int_{\left(\mathcal{M}, g_0\right)} v^{\frac{2 n}{n-2}-k \frac{n+2}{n-2}} \sigma_k(\lambda(V))-\frac{n-2}{2n} \int_{\left(\mathcal{M}, g_0\right)} v^{\frac{2n}{n-2}} . \end{aligned} $$