the obstacle, namely, $$\label{e301}
   \mathcal{S}[\varphi,u_0]=\left\{v\in C(\overline\Omega)\,:\,v \mbox{ convex }, v|_{\partial\Omega}=\varphi, Dv(\Omega)\subset D\varphi(\overline\Omega), v\geq u_0\mbox{ in }\Omega\right\},$$ where $\varphi$ is a smooth, uniformly convex function defined on a neighborhood of $\overline\Omega$.

\[l301\] Suppose $0\leq \alpha\leq\frac{1}{n+2}$. There exists a sequence of functions $\{u_i\}$ in $\mathcal{S}[\varphi, u_0]$ such that each $u_i$ is the maximizer of the functional $$J^i_{\alpha}(v)=J_\alpha(v)-\int_\Omega G_i(x,v), \ v\in \mathcal{S}[\varphi, u_0]$$ and $u_i\to u_0$ as $i\to \infty$, where $G_i(x,t)$ is a smooth, convex function monotone decreasing in $t$ . Furthermore, there is no obstacle for $u_i$ in $\Omega$, i.e., $u_i(x)>u_0(x)$, $x\in \Omega$.

First, we consider a penalized problem. The idea is inspired by [@STW]. Define $$\label{e302}
J_{\alpha, g}(v)=J_\alpha(v)-\int_\Omega G(x,v),$$ where $G(x,t)$ is a smooth, convex function monotone decreasing in $t$ such that $$\label{e303}
     G(x,t)\geq a(x)(t-u_0(x))^{-n}\quad\mbox{for }t>u_0(x), \ x \in \Omega.$$ Here $a$ is a positive function in $\Omega$, with $a(x)\to 0$ fast enough as $x\to\partial\Omega$ such that the set $\{v\in \mathcal{S}[\varphi, u_0]\,:\,J_{\alpha, g}(v)>-\infty\}\neq\emptyset$. It is clear that $J_{\alpha,g}$ is still concave, upper semi-continuous and bounded from above. Hence there is a unique maximizer $v_g$ to the problem $$\label{e304}
   \sup\{J_{\alpha, g}(v)\,:\,v\in \mathcal{S}[\varphi, u_0]\}.$$

We claim that for any $x\in\Omega$, $$\label{e305}
        v_g(x)>u_0(x).$$ Indeed, if there is a point $x_0\in\Omega$ such that $v_g(x_0)=u_0(x_0)$, by 