Uniform estimate of a semilinear elliptic equation with Robin boundary condition.

Question

Consider $u\in C^2(\Omega )\cap C(\bar \Omega )(\Omega \subset \mathbb{R}^d)$ which satisfies the following equations \begin{cases} -\Delta u+|u |^ru=f &\text{ in }\Omega,\\ \frac{\partial u }{\partial n }+\alpha u=g & \text{ on }\partial \Omega, \end{cases} where $\alpha(x)>\alpha_0>0$ and $f\in L^\infty(\Omega ),g\in L^\infty( \partial \Omega )$. I want to obtain a uniform estimate of $u$ as follows: $$\sup_{\bar \Omega }\le \frac 1{\alpha_0}\sup_{\partial \Omega }|g|+ \sup_{\Omega}|f|^{\frac 1{1+r}} .$$ My idea is to construct auxiliary function $v(x)$ so that $w(x)=u(x)-v(x)$ satisfies $ -\Delta w\le 0$ and $ \partial w/\partial n+\alpha w|_{\partial \Omega}\le 0$, which implies $w\le 0$ by maximum principle. However, the construction of $v(x)$ is just a bit difficult for me: I tried the following two formation of $v(x)$: Denote $G=\sup_{\partial \Omega}|g|, F=\sup_{\Omega }|f|, l=diam(\Omega)$ and let $$v(x)=\frac{ G}{\alpha_0} +\frac{ F^{\frac 1{r+1} }}{2d}\left(\frac{l^2+1}{\alpha_0}+l^2-|x|^2 \right).$$ But it seems that $-\Delta w\le 0$ does not hold and thus I cannot go further. Can anyone give a proper construction of $v(x)$? Detailed proof will be appreciated:)