Non-existence of extremals for the Adimurthi–Druet inequality

The Adimurthi–Druet [1] inequality is an improvement of the standard Moser–Trudinger inequality by adding a L2-type perturbation, quantified by α∈[0,λ1), where λ1 is the first Dirichlet eigenvalue of Δ on a smooth bounded domain. It is known [3,10,14,19] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter α approaches λ1. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as α→λ1.