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In dimension two, we investigate a free energy and the ground state energy ofthe Schr\"odinger-Poisson system coupled with a logarithmic nonlinearity interms of underlying functional inequalities which take into account the scalinginvariances of the problem. Such a system can be considered as a nonlinearSchr\"odinger equation with a cubic but nonlocal Poisson nonlinearity, and alocal logarithmic nonlinearity. Both cases of repulsive and attractive forcesare considered. We also assume that there is an external potential with minimalgrowth at infinity, which turns out to have a logarithmic growth. Our estimatesrely on new logarithmic interpolation inequalities which combine logarithmicHardy-Littlewood-Sobolev and logarithmic Sobolev inequalities. Thetwo-dimensional model appears as a limit case of more classical problems inhigher dimensions.

Logarithmic estimates for mean-field models in dimension two and the Schr\"odinger-Poisson system