Extremal functions for the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}(\mathbb R^N)$

TL;DR

This paper investigates the existence of maximizers for a Moser--Trudinger inequality of Adimurthi--Druet type in \\mathbb{R}^N, establishing conditions for attainability in subcritical and critical cases, and analyzing blow-up behavior.

Contribution

It provides new results on the attainability of extremal functions for the inequality in unbounded domains, including critical and subcritical cases, and addresses open questions from prior work.

Findings

Maximizers exist in subcritical case for N≥3 or N=2 with specific beta range.

No maximizers for small beta in 2D case.

Attainability in critical case for small alpha.

Abstract

We study the existence and nonexistence of maximizers for variational problem concerning to the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}(\mathbb R^N)$ \[ MT(N,\beta, \alpha) =\sup_{u\in W^{1,N}(\mathbb R^N), \|\nabla u\|_N^N + \|u\|_N^N\leq 1} \int_{\mathbb R^N} \Phi_N(\beta(1+\alpha \|u\|_N^N)^{\frac1{N-1}} |u|^{\frac N{N-1}}) dx, \] where $\Phi_N(t) =e^{t} -\sum_{k=0}^{N-2} \frac{t^k}{k!}$, $0\leq \alpha < 1$ both in the subcritical case $\beta < \beta_N$ and critical case $\beta =\beta_N$ with $\beta_N = N \omega_{N-1}^{\frac1{N-1}}$ and $\omega_{N-1}$ denotes the surface area of the unit sphere in $\mathbb R^N$. We will show that $MT(N,\beta,\alpha)$ is attained in the subcritical case if $N\geq 3$ or $N=2$ and $\beta \in (\frac{2(1+2\alpha)}{(1+\alpha)^2 B_2},\beta_2)$ with $B_2$ is the best constant in a Gagliardo--Nirenberg inequality in $W^{1,2}(\mathbb R^2)$. We also show that $MT(2,\beta,\alpha)$ is not attained for $\beta$ small which is different from the context of bounded domains. In the critical case, we prove that $MT(N,\beta_N,\alpha)$ is attained for $\alpha\geq 0$ small enough. To prove our results, we first establish a lower bound for $MT(N,\beta,\alpha)$ which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of $MT(N,\beta,\alpha)$ in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together the scaling argument, we show that $MT(N,\beta_N,1) =\infty$. Our results settle the questions left open in \cite{doO2015,doO2016}.