Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy 
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x),   \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function. 

I would like to know how the solutions depend on $\alpha$. Is $u$ a continuous of differentiable function with respect to $\alpha$? I will also appreciate any reference.