The following nonlinear elliptic PDE arose in my research:
$$\Delta f - e^f \partial_s f = E(s,t)\,,$$
where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the unknown function, $E(s,t)$ is a given *nonnegative* function which decays exponentially together with all derivatives as $|s| \to \infty$, and the Laplacian is $\Delta = \partial_s^2 + \partial_t^2$; $f$ is required to satisfy the following asymptotic conditions:
$$\lim_{s \to \pm \infty} f(s,t) = a_\pm \in \mathbb R\,,$$
for some given constants $a_\pm$; it can be seen using integration over $s,t$ that these must satisfy
$$e^{a-} - e^{a_+} = \int_{\mathbb R \times \mathbb R/ \mathbb Z}E(s,t)\,ds\,dt\,.$$

**Question:** *does there always exist a unique solution of this equation for $f$ satisfying the given asymptotic conditions?*

I checked standard sources such as Gilbarg and Trudinger, but even uniqueness doesn't seem to follow immediately from the form of the eqation, even though the maximum principle *does hold*.