Existence and non-existence of maximizers for the Moser-Trudinger type inequalities under inhomogeneous constraints

Abstract

In this paper, we study the existence and non-existence of maximizers for the Moser-Trudinger type inequalities in $\Bbb R^N$ of the form \[ D_{N,α}(a,b):= \sup_{u\in W^{1,N}(\Bbb R^N),\,\|\nabla u\|_{L^N(\Bbb R^N)}^a+\|u\|_{L^N(\Bbb R^N)}^b=1} \int_{\Bbb R^N}Φ_N\left(α|u|^{N'}\right)dx. \] Here $N\geq 2$, $N'=\frac{N}{N-1}$, $a,b>0$, $α\in (0,α_N]$ and $Φ_N(t):=e^t-\sum_{j=0}^{N-2}\frac{t^j}{j!}$ where $α_N:= N ω_{N-1}^{1/(N-1)}$ and $ω_{N-1}$ denotes the surface area of the unit ball in $\Bbb R^N$. We show the existence of the threshold $α_\ast = α_\ast(a,b,N) \in [0,α_N]$ such that $D_{N,α}(a,b)$ is not attained if $α\in (0,α_\ast)$ and is attained if $ α\in (α_\ast , α_N)$. We also provide the conditions on $(a,b)$ in order that the inequality $α_\ast < α_N$ holds.