Equivalent Moser type inequalities in R2 and the zero mass case

We first investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality is equivalent to an improved version of the subcritical Adachi-Tanaka inequality which we prove to be attained. Then, we consider smooth compactly supported functions with respect to the Dirichlet norm {norm of matrix}∇,{dot operator}{norm of matrix}2, and we prove an optimal Lorentz-Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in H1(R2) such as the Adachi-Tanaka inequality and a version of Ruf's inequality.