On Trudinger-Moser type inequalities with logarithmic weights

Trudinger–Moser type inequalities for radial Sobolev spaces with logarithmic weights are considered. The precise Trudinger–Moser growths in dependence on the logarithmic terms, and the corresponding sharp Moser type exponents are determined. In a particular case a critical Trudinger–Moser growth of double exponential type is found.