Non-existence of extremals for the Adimurthi-Druet inequality

TL;DR

This paper proves that the Adimurthi-Druet inequality lacks extremal functions as the perturbation parameter approaches the first eigenvalue, revealing a singularity in the problem.

Contribution

It demonstrates the non-existence of extremals for the Adimurthi-Druet inequality near the critical perturbation parameter, extending understanding of the inequality's extremal behavior.

Findings

Extremals exist for small perturbation parameters.

No extremals exist as the parameter approaches the first eigenvalue.

Sharp energy expansions characterize the singularity at the critical parameter.

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 $\alpha\to \lambda\_1$.