Non-existence of extremals for the Adimurthi-Druet inequality

Abstract

The Adimurthi-Druet [1] inequality is an improvement of the standard Moser-Trudinger inequality by adding a $L^2$-type perturbation, quantified by $\alpha \in [0,\lambda \_1)$, where $\lambda \_1$ is the first Dirichlet eigenvalue of $\Delta$ on a smooth bounded domain. It is known [3,9,13,18] that this inequality admits extremal functions, when the perturbation parameter $\alpha$ is small. By contrast, we prove here that the Adimurthi-Druet inequality does not admit any extremal, when the perturbation parameter $\alpha$ approaches $\lambda \_1$. 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 $\alpha \to \lambda \_1$.

References in corpus (2)

• The Lin-Ni's problem for mean convex domains
• On Lin-Ni's conjecture in dimensions four and six

Cited by in corpus (3)

• Supercritical Moser-Trudinger inequalities and related elliptic problems
• A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems
• A remark on energy estimates concerning extremals for Trudinger-Moser inequalities on a disc