Schrödinger–Poisson systems with zero mass in the Sobolev limiting case

We study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted $N$-Laplace operator and without the mass term, and a higher-order fractional Poisson equation. Since the system is set in $R^N$, 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 logarithm, hence sign-changing and unbounded from above and below. 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. Our results are new even in the planar case $N=2$.