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Trudinger-Moser inequality on the whole plane with the exact growth condition

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^\infty$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modified versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrodinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.