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

Van Hoang Nguyen

Published 2017-02-26Version 1

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 $\Phi_N(t) =e^{t} -\sum_{k=0}^{N-2} \frac{t^k}{k!}$, $0\leq \alpha < 1$ and $\beta \leq \beta_N = N \omega_{N-1}^{\frac1{N-1}}$ with $\omega_{N-1}$ denotes the surface are of the unit sphere in $\mathbb R^N$. We will show that $MT(N,\beta,\alpha)$ is attained for $\beta \in (0,\beta_N)$ if $N\geq 3$ and for $\beta \in (\frac{2(1+2\alpha)}{(1+\alpha)^2 B_2},\beta_2)$ if $N = 2$ with $B_2$ defined by \eqref{eq:B2def} is the best constant in a Gagliardo--Nirenberg inequality in $W^{1,2}(\mathbb R^2)$. We also prove that there exists $\alpha_0\in (0,1)$ such that $MT(N,\beta_N,\alpha)$ is attained for any $0\leq \alpha < \alpha_0$. Finally, we prove that $MT(2,\beta,\alpha)$ is not attained for $\beta$ small which is different from the bounded domain case. Our proof is based on the careful estimate of the maximizing level with the aid of normalized vanishing sequence in the subcritical case (i.e., $\beta < \beta_N$), and on the blow-up analysis method in the critical case (i.e., $\beta =\beta_N$). 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}.