An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W1,n(Rn)W1,n(Rn), n⩾2n⩾2, into the Orlicz space LΦαLΦα determined by the Young function Φα(s)Φα(s) behaving like eα|s|n/(n−1)−1eα|s|n/(n−1)−1 as |s|→+∞|s|→+∞. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space RnRn.