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 ${\mathbb{R}}^N$ 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 $N\ge 2$, $N^{\prime }=\frac{N}{N-1}$, $a,b>0$, $\alpha \in (0,{\alpha }_N]$ and ${\Phi }_N(t):=e^t-{\sum }_{j=0}^{N-2}\frac{t^j}{j!}$ where ${\alpha }_N:=N{\omega }_{N-1}^{1/(N-1)}$ and ${\omega }_{N-1}$ denotes the surface area of the unit ball in ${\mathbb{R}}^N$. We show the existence of the threshold ${\alpha }_{\ast }={\alpha }_{\ast }(a,b,N)\in [0,{\alpha }_N]$ such that $D_{N,\alpha }(a,b)$ is not attained if $\alpha \in (0,{\alpha }_{\ast })$ and is attained if $\alpha \in ({\alpha }_{\ast },{\alpha }_N)$. We also provide the conditions on $(a,b)$ in order that the inequality ${\alpha }_{\ast }<{\alpha }_N$ holds.

Mathematische Annalen, 2019