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The paper raises a question about the optimal critical nonlinearity for theSobolev space in two dimensions, connected to loss of compactness, anddiscusses the pertinent concentration compactness framework. We studyproperties of the improved version of the Trudinger-Moser inequality on theopen unit disk $B\subset\R^2$, recently proved by G. Mancini and K. Sandeep.Unlike the original Trudinger-Moser inequality, this inequality is invariantwith respect to M\"obius automorphisms of the unit disk, and as such is acloser analogy of the critical nonlinearity $\int |u|^{2^*}$ in the higherdimension than the original Trudinger-Moser nonlinearity.

On a version of Trudinger-Moser inequality with M\"obius shift invariance