is the class of functions $$\label{e103}
   \mathcal{S}[\varphi,\psi]=\left\{u\in C(\overline\Omega)\,:\,u \mbox{ convex }, u|_{\partial\Omega}=\varphi, Du(\Omega)\subset D\varphi(\overline\Omega), u\geq\psi \mbox{ in }\Omega\right\},$$ $\varphi$ is a smooth, uniformly convex function defined on a neighborhood of $\overline\Omega$, $\psi$ is an obstacle function, and $Du(\Omega)$ represents the image of the subgradients of $u$ at all points $x\in\Omega$.

The Euler-Lagrange equations of are a class of fourth order equations, that is, $$\label{e104}
   U^{ij}w_{ij}=f,$$ where $(U^{ij})$ is the cofactor matrix of the Hessian $D^2u$, and $$\label{e105}
   w=\left[{\mbox{det}}\,D^2u\right]^{-(1-\alpha)},\ \ \alpha\geq 0.$$ When $\alpha=\frac{1}{n+2}$, equation is the affine mean curvature equation and the functional is the affine area functional. When $\alpha=0$, equation is Abreu’s equation arising from the study of Calabi’s extremal metrics on toric Kähler manifolds [@D1; @D2; @D3; @D4].

Due to their importance in geometry, variational problems of have attracted much interest in recent years. In the case of $\alpha=\frac{1}{n+2}$, the variational problem without obstacle is the graph case of affine Plateau problem [@TW08; @TW10], raised by Calabi and Chern. The case of $\alpha=0$ has been treated in [@Z1]. The obstacle problem of affine maximal surfaces was first introduced in [@STW]. In this paper, we obtain:

\[t101\] Suppose $n=2$, $0\leq\alpha\leq \frac{1}{n+2}$, and $f\in L^\infty(\Omega)$. Let $\varphi$ be a smooth, 