Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case

We study the nonlinear Schrödinger equation for the s-fractional p-Laplacian strongly coupled with the Poisson equation in dimension two and with p=2s, which is the limiting case for the embedding of the fractional Sobolev space Ws,p(R2). We prove existence of solutions by means of a variational approximating procedure for an auxiliary Choquard equation in which the uniformly approximated sign-changing logarithmic kernel competes with the exponential nonlinearity. Qualitative properties of solutions such as symmetry and decay are also established by exploiting a suitable moving planes technique.