Vanishing-concentration-compactness alternative for critical Sobolev embedding with a general integrand in R2

A maximizing problem associated with the Moser–Pohozaev–Trudinger–Yudovich type inequality proved in Ruf (J Funct Anal 219:340–367, 2005) states: d ( α ) : = sup u ∈ H 1 ( R 2 ) , ‖ u ‖ H 1 = 1 ∫ R 2 ( e α | u | 2 - 1 ) < ∞ for 0 < α ≤ 4 π . Do Ó–Sani–Tarsi (Commun Contemp Math 20:1650036, 2018) established the vanishing-concentration-compactness (VCC) alternative with respect to any maximizing sequence for d ( α ) . In this paper, we consider a maximizing problem with a general integrand: d Φ ( α ) : = sup u ∈ H 1 ( R 2 ) , ‖ u ‖ H 1 = 1 ∫ R 2 Φ α ( | u | ) , where Φ ∈ C ( [ 0 , ∞ ) ) with α > 0 and Φ α ( s ) : = Φ ( α s ) for s ≥ 0 . Our aim is to extract sufficient conditions for Φ under which the (VCC) alternative still holds for d Φ ( α ) . As a result, assuming lim s → ∞ Φ ( s ) Φ ( ( 1 + ε ) s ) = 0 with any small ε > 0 as an essential condition, we prove the (VCC) alternative for d Φ ( α ) which extends not only d ( α ) but other various examples untreated so far.