arXiv2018
We study the existence and nonexistence of maximizers for variational problem concerning to the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}(\mathbb R^N)$ \[ MT(N,β, α) =\sup_{u\in W^{1,N}(\mathbb R^N), \|\nabla u\|_N^N + \|u\|_N^N\leq 1} \int_{\mathbb R^N} Φ_N(β(1+α\|u\|_N^N)^{\frac1{N-1}} |u|^{\frac N{N-1}}) dx, \] 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}.