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 $L^2$-type perturbation, quantified by $α\in [0,λ\_1)$, where $λ\_1$ is the first Dirichlet eigenvalue of $Δ$ on a smooth bounded domain. It is known [3,9,13,18] 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 $λ\_1$. 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 $α\to λ\_1$.

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