Trudinger–Moser‐type inequality with logarithmic convolution potentials

We establish Moser-Trudinger type inequalities in presence of a logarithmic convolution potential when the domain is a ball or the entire space $R^2$. Moreover, we characterize critical nonlinear growth rates for these inequalities to hold and for the existence of corresponding extremal functions. In addition, we show that extremal functions satisfy corresponding Euler-Lagrange equations, and we derive general symmetry and uniqueness results for solutions of these equations.