But this is not true for other $\alpha$. It has been proved in [@Z1] that for $\alpha=0$, under the rotations in $\mathbb{R}^{n+1}$, equation changes in a proper way such that the determinant estimate in Lemma \[l202\] still holds.

For our purpose, we consider a more general functional $$\label{e215}
        J_\alpha(u)=A_\alpha(u)-\int_\Omega F(x,u)dx,$$ where $A_\alpha$ is in , $F(x, t)$ is a function on $\Omega\times \mathbb R$. Let $u$ be a locally critical point of the functional $J_\alpha$, thus it satisfies with the inhomogeneous term $f=F_t:=\frac{\partial F}{\partial t}$.

Consider the rotation $Z=TX$, given by $z_1=-x_{n+1}, z_{n+1}=x_1$, $z_i=x_i$ for $2\leq i\leq n$. Assume the graph of $u$, $\mathcal{G}_u=\{(x,u(x))\,:\,x\in\Omega\}$, can be represented by a convex function $z_{n+1}=v(z_1,\cdots,z_n)$ in $z$-coordinates over a domain $\hat\Omega$. Following the computation in [@Z2], $v$ is a locally critical point of $$\label{e216}
\hat J_\alpha(v)=\hat A_\alpha(v)-\int_{\hat\Omega}F(v,z_2,\cdots,z_n,-z_1),$$ where $$\hat A_\alpha(v)=
\begin{cases}
 \int_{\hat \Omega}[{\mbox{det}}D^2v]^\alpha|v_1|^{1-(n+2)\alpha}\,dz, & \alpha>0,\\[5pt]
\int_{\hat \Omega} [\log{\mbox{det}}D^2v 
-\frac{n+2}{2}\log(v_1^2)] (v_1^2)^{\frac{1}{2}}\,dz, &\ \alpha=0.
\end{cases}$$ When $\alpha>0$, by computing the Euler equation, we can obtain the corresponding equation for $v$, that is, $$\label{e217}
    \begin{split}
        \alpha v_1^{1-\alpha(n+2)}V^{ij}(d^{\alpha-1})_{ij}&+(1-\alpha)\alpha(n+2)(1-\alpha(n+2))v_1^{-\alpha(n+2)-1}v_{11}d^\alpha \\
            &+(1-\alpha(n+2))(2\alpha-2)v_1^{-\alpha(n+2)}(d^\alpha)_1= F_t,
    \end{split}$$ or equivalently, denoting $\lambda = 1-\alpha(n+2)$, $$\label{e218}
        V^{ij}(d^{\alpha-1})_{ij}=g+\mathcal{F}_t,$$ where $(V^{ij})$ is the cofactor matrix of $(v_{ij})$, $d={\mbox{det}}\,D^2v$ and $$\begin{aligned}
        g\!\!&=&\!\!2\lambda(1-\alpha)d^\alpha v^{ij}v_{ij1}\frac{1}{v_1}-(1-\alpha)(n+2)\lambda d^\alpha\frac{v_{11}}{v_1^2}, \\
        \mathcal{F}_t\!\!&=&\!\!\alpha^{-1} \frac{F_t}{v_1^\lambda}, 