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

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

Automated summary

In two dimensions, we study the free energy and ground state energy of the Schrodinger-Poisson system coupled with a logarithmic nonlinearity, taking into account scaling invariances, external potential with minimal growth, and new logarithmic interpolation inequalities. This two-dimensional model serves as a limit case of more classical problems in higher dimensions, and considers both repulsive and attractive forces.