Nonlinear Schr\"odinger-Poisson systems in dimension two: the zero mass case

We provide an existence result for a Schr\"odinger-Poisson system in gradient form, set in the whole plane, in the case of zero mass. Since the setting is limiting for the Sobolev embedding, we admit nonlinearities with subcritical or critical growth in the sense of Trudinger-Moser. In particular, the absence of the mass term requires a nonstandard functional framework, based on homogeneous Sobolev spaces. These features, combined with the logarithmic behaviour of the kernel of the Poisson equation, make the analysis delicate, since standard variational tools cannot be applied. The system is solved by considering the corresponding logarithmic Choquard equation. The existence of a mountain pass-type solution is established by means of a careful analysis of appropriate Cerami sequences, whose boundedness is ensured through a nonstandard variational method, suggested by the subtle nature of the functional geometry involved. As a key tool in our estimates, we also introduce a logarithmic weighted Trudinger-Moser inequality, along with a related Cao-type inequality, both of which hold in our functional setting and are, we believe, of independent interest.