A class of Hartree-Fock systems with null mass via Nehari-Pohozaev with logarithmic interactions

We establish the existence and qualitative properties of nontrivial solutions for a class of Hartree-Fock type systems defined over the whole space ${\mathbb{R}}^2$. By introducing a suitable Nehari-Pohozaev manifold, we prove the existence, regularity and we describe the asymptotic behavior of solutions with respect to the interaction parameter $\beta >0$. In particular, we show that the system admits either a vector ground state or a semitrivial ground state solution, depending on the magnitude of $\beta$.