The following nonlinear elliptic PDE arose in my research: $$\Delta f - e^f \partial_s f = E(s,t)\,,$$ where $f : \mathbb R \times \mathbb R/\mathbb Z \to \mathbb R$, $f = f(s,t)$, is the unknown function, $E(s,t)$ is a given nonnegative function which decays exponentially together with all derivatives as $|s| \to \infty$; $f$ 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.

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.