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 or differentiable function with respect to $\alpha$? I will also appreciate any reference.