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

Van Hoang Nguyen|arXiv (Cornell University)|Feb 26, 2017

Advanced Mathematical Physics Problems32 references5 citations

TL;DR

This paper establishes the existence and nonexistence of extremal functions for the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}( r^N)$, proving attainability in the subcritical case for $N \geq 3$ or $N=2$ under specific $β$-range, and in the critical case for small $\alpha$. It further shows $MT(N,\beta_N,1) = \infty$, resolving open questions on extremal function behavior in unbounded domains using blow-up analysis and Moser sequence scaling.

ABSTRACT

We study the existence and nonexistence of maximizers for variational problem concerning to the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}(\mathbb R^N)$ \[ MT(N,β, α) =\sup_{u\in W^{1,N}(\mathbb R^N), \| abla u\|_N^N + \|u\|_N^N\leq 1} \int_{\mathbb R^N} Φ_N(β(1+α\|u\|_N^N)^{\frac1{N-1}} |u|^{\frac N{N-1}}) dx, \] where $Φ_N(t) =e^{t} -\sum_{k=0}^{N-2} \frac{t^k}{k!}$, $0\leq α< 1$ both in the subcritical case $β< β_N$ and critical case $β=β_N$ with $β_N = N ω_{N-1}^{\frac1{N-1}}$ and $ω_{N-1}$ denotes the surface area of the unit sphere in $\mathbb R^N$. We will show that $MT(N,β,α)$ is attained in the subcritical case if $N\geq 3$ or $N=2$ and $β\in (\frac{2(1+2α)}{(1+α)^2 B_2},β_2)$ with $B_2$ is the best constant in a Gagliardo--Nirenberg inequality in $W^{1,2}(\mathbb R^2)$. We also show that $MT(2,β,α)$ is not attained for $β$ small which is different from the context of bounded domains. In the critical case, we prove that $MT(N,β_N,α)$ is attained for $α\geq 0$ small enough. To prove our results, we first establish a lower bound for $MT(N,β,α)$ which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of $MT(N,β,α)$ 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 $MT(N,β_N,1) =\infty$. Our results settle the questions left open in \cite{doO2015,doO2016}.

Motivation & Objective

• To resolve open questions on the existence and nonexistence of maximizers for the Moser--Trudinger inequality of Adimurthi--Druet type in unbounded domains.
• To determine the conditions under which the supremum $MT(N,\beta,\alpha)$ is attained in $W^{1,N}(\mathbb{R}^N)$ for both subcritical ($\beta < \beta_N$) and critical ($\beta = \beta_N$) cases.
• To clarify the role of the parameter $\alpha \in [0,1)$ in the critical and subcritical regimes, particularly regarding concentration and vanishing of maximizer sequences.
• To establish sharp asymptotic behavior and blow-up profiles for extremal sequences via refined analysis.
• To prove that $MT(N,\beta_N,1) = \infty$, showing the failure of attainability at the endpoint $\alpha=1$.

Proposed method

• Establishing a lower bound for $MT(N,\beta,\alpha)$ to rule out vanishing or concentrating behavior in maximizer sequences, ensuring precompactness in the subcritical case.
• Applying blow-up analysis to study the concentration profiles of maximizing sequences in the critical case $\beta = \beta_N$, identifying the blow-up point and profile.
• Using the Moser sequence and scaling arguments to construct test functions that demonstrate unboundedness of $MT(N,\beta_N,1)$.
• Employing the transformation between radial functions on $\mathbb{R}^N$ and functions on $\mathbb{R}$ via $w(t) = N^{1-1/N}\omega_{N-1}^{1/N}u(e^{-t/N})$ to relate integrals to one-dimensional weighted integrals.
• Applying integration by parts and asymptotic estimates on $\Phi_N(t) = e^t - \sum_{k=0}^{N-2} \frac{t^k}{k!}$ to control the growth of the exponential functional.
• Using the Gagliardo--Nirenberg inequality to determine the threshold $\beta$-range for $N=2$ in the subcritical case.

Experimental results

Research questions

• RQ1 Under what conditions is the supremum $MT(N,\beta,\alpha)$ attained in $W^{1,N}(\mathbb{R}^N)$ for $\beta < \beta_N$?
• RQ2 Is $MT(2,\beta,\alpha)$ attained for small $\beta$, and how does this differ from the bounded domain case?
• RQ3 Does $MT(N,\beta_N,\alpha)$ remain finite and attainable for $\alpha > 0$ small in the critical case?
• RQ4 What happens to $MT(N,\beta_N,\alpha)$ as $\alpha \to 1^-$, particularly at $\alpha=1$?
• RQ5 Can the blow-up analysis method be used to characterize the concentration profile of maximizers in the critical case?

Key findings

• The supremum $MT(N,\beta,\alpha)$ is attained in the subcritical case for $N \geq 3$ and for $N=2$ when $\beta \in \left(\frac{2(1+2\alpha)}{(1+\alpha)^2 B_2}, \beta_2\right)$, where $B_2$ is the best constant in the Gagliardo--Nirenberg inequality in $W^{1,2}(\mathbb{R}^2)$.
• For $N=2$, $MT(2,\beta,\alpha)$ is not attained when $\beta$ is sufficiently small, contrasting with the bounded domain case.
• In the critical case $\beta = \beta_N$, $MT(N,\beta_N,\alpha)$ is attained for all $\alpha \geq 0$ small enough.
• The paper proves $MT(N,\beta_N,1) = \infty$ using the Moser sequence and scaling, showing that the functional becomes unbounded at $\alpha=1$, which settles an open problem.
• The blow-up analysis method successfully identifies the concentration profile and allows the construction of a maximizer in the critical case for small $\alpha$.
• A lower bound for $MT(N,\beta,\alpha)$ is established that excludes vanishing and concentrating behaviors, ensuring the existence of a maximizer in the subcritical regime.