  We consider the $N$-Laplacian Schr\"odinger equation strongly coupled with
higher order fractional Poisson's equations. When the order of the Riesz
potential $\alpha$ is equal to the Euclidean dimension $N$, and thus it is a
logarithm, the system turns out to be equivalent to a nonlocal Choquard type
equation. On the one hand, the natural function space setting in which the
Schr\"odinger energy is well defined is the Sobolev limiting space
$W^{1,N}(\mathbb{R}^N)$, where the maximal nonlinear growth is of exponential
type. On the other hand, in order to have the nonlocal energy well defined and
prove the existence of finite energy solutions, we introduce a suitable
$log$-weighted variant of the Pohozaev-Trudinger inequality which provides a
proper functional framework where we use variational methods.
