Trudinger–Moser‐type inequality with logarithmic convolution potentials

We establish Moser–Trudinger-type inequalities in the presence of a logarithmic convolution potential when the domain is a ball or the entire space R 2 R² . 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.