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 \[ D_{N,\alpha}(a,b):= \sup_{u\in W^{1,N}(\Bbb R^N),\,\|\nabla u\|_{L^N(\Bbb R^N)}^a+\|u\|_{L^N(\Bbb R^N)}^b=1} \int_{\Bbb R^N}\Phi_N\left(\alpha|u|^{N'}\right)dx. \] Here $N\geq 2$, $N'=\frac{N}{N-1}$, $a,b>0$, $\alpha \in (0,\alpha_N]$ 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 $\Bbb R^N$. 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.