Some sharp inequalities related to Trudinger-Moser inequality

This paper deals with improvements of the Trudinger-Moser inequality related to the operator QV(u):=−Δnu+V(x)|u|n−2u, where n≥2 and the potential V:Rn→R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E:={u∈W1,n(Rn):∫RnV(x)|u|ndx<∞} endowed with the norm ‖u‖:=[∫Rn(|∇u|n+V(x)|u|n)dx]1/n and we prove that if (uk) is a sequence in E such that ‖uk‖=1, uk⇀u≢0 in E and 0