Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system

In dimension two, we investigate a free energy and the ground state of Schrödinger–Poisson system coupled with logarithmic nonlinearity in terms underlying functional inequalities which take into account scaling invariances problem. Such can be considered as nonlinear Schrödinger equation cubic but nonlocal Poisson nonlinearity, local nonlinearity. Both cases repulsive attractive forces are considered. We also assume that there is an external potential minimal growth at infinity, turns out to have growth. Our estimates rely on new interpolation combine Hardy–Littlewood–Sobolev Sobolev inequalities. The two-dimensional model appears limit case more classical problems higher dimensions.