  We study the existence of positive solutions for a class of systems which
strongly couple a quasilinear Schr\"odinger equation driven by a weighted
$N$-Laplace operator and without the mass term, and a higher-order fractional
Poisson equation. Since the system is considered in $\mathbb R^N$, the limiting
case for the Sobolev embedding, we consider nonlinearities with exponential
growth. Existence is proved relying on the study of a corresponding Choquard
equation in which the Riesz kernel is a sign-changing logarithm. This is in
turn solved by means of a variational approximating procedure for an auxiliary
Choquard equation where the logarithm is uniformly approximated by polynomial
kernels.
