A (nontrivial) nonlinear Poisson equation Let $U \subset \mathbb R^3$ be bounded, with smooth boundary. Does the equation
$$ -\Delta f = e^{-f} \text{ in } U, \quad f = 0 \text{ on } \partial U$$
has a solution? Is there a name for this equation? Any idea or hint or reference is highly appreciated.

I don't have a definitive answer, but this makes it a little more tractable. The differential operator is uniquely invertible and has a Green's function $G(x,y)$ that is positive. This allows us to reformulate our equation as $$f(x) = \int_U G(x,y)e^{-f(y)}dy = \Phi(f).$$ If you can show that the integral operator on the RHS is a contraction, then the solution exists and is unique by the fixed point theorem. 
Let us use the $\infty$-norm for this analysis. Then we have
$$\|\Phi(f)-\Phi(g)\| \leq \int_U\|G(\cdot,y)\|\|e^{-f(y)}-e^{-g(y)}\|dy.$$
Now notice that any solution of this integral equation must be positive, so the exponentials are both bounded above by 1. Now we have
$$\|\Phi(f)-\Phi(g)\| \leq \int_U\|G(\cdot,y)\|dy.$$
If this value can be bounded above by 1, then there is a unique solution. However, this depends (only) on the domain. This value can also be bounded (in a different norm, so you must be more careful), by the reciprocal of the smallest eigenvalue of the Laplace operator on this domain, which (I believe) is generally inversely proportionally to the volume of the region $U$. Therefore, we expect this problem to have a unique solution on small domains rather than large domains.
