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

The Trudinger-Moser inequality is a substitute for the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L^∞ . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails to hold on the whole plane, but a few modified versions are available. We prove a more precise version of the latter, giving necessary and sufficient conditions for boundedness, as well as for compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrödinger 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.