A Kazhdan-Warner type problem

Let $X$ be a compact Riemannian manifold and I am interested in the following set of equations:
begin{align*}
Delta f+ucdot e^{f+lambda}=c\
lambda-2f=g
end{align*}

where $u,g$ are given real valued functions on $X$. $u>0$ and $c$ is a constant. We need to solve for $f$ and $lambda$. Notice that for a fixed $lambda$ the first equation always has a solution if $c>0$. So, we can put that in our assumption as well. Is there any conditions which assures solutions to these set of equations? If yes then is it unique? Any source, comment or ideas are welcome.