Quasilinear logarithmic choquard equations with exponential growth in RN

We consider the N-Laplacian Schrödinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential α 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 Schrödinger energy is well defined is the Sobolev limiting space W1,N(RN), 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 suitable functional framework where we use variational methods.