Vanishing-concentration-compactness alternative for the Trudinger-Moser inequality in R^N

Let (Formula presented.), (Formula presented.) and (Formula presented.). Our aim is to clarify the influence of the constraint (Formula presented.) on concentration phenomena of (spherically symmetric and non-increasing) maximizing sequences for the Trudinger–Moser supremum (Formula presented.) where (Formula presented.) is the sharp exponent of Moser, i.e. (Formula presented.) and (Formula presented.) is the surface measure of the (Formula presented.)-dimensional unit sphere in (Formula presented.). We obtain a vanishing-concentration-compactness alternative showing that maximizing sequences for (Formula presented.) cannot concentrate either when (Formula presented.) or when (Formula presented.) and (Formula presented.) is sufficiently small. From this alternative, we deduce the attainability of (Formula presented.) for special values of the parameters (Formula presented.) and (Formula presented.).