Abstract: In this paper, we investigate the minimization problem $$\begin{equation*} e_{s}(\rho)=\inf_{u\in W^{1,N}_{V}(\mathbb{R}^{N}),\|u\|^{N}_{N}=\rho>0}E(u), \end{equation*}$$ where $$E(u)=\frac{1}{N}\int_{\mathbb{R}^{N}}|\nabla u|^{N}{\rm d}x+\frac{1}{N}\int_{\mathbb{R}^{N}}V(x)|u|^{N}{\rm d}x-\frac{1}{s}\int_{\mathbb{R}^{N}}|u|^{s}{\rm d}x.$$ Here $s>N$, $V$ is a spherically symmetric increasing function satisfying $$V(0)=0, \lim_{|x|\rightarrow\infty}V(x)=+\infty.$$ We discuss the problem in three cases. First, for the case $s>2N$, $e_{s}(\rho)=-\infty$ for any $\rho>0$. Secondly, for the case $N<s<2N$, for any $\rho>0$, we prove that it admits a minimizer which is nonnegative, spherically symmetric and decreasing via the $N$-Laplacian Gagliardo-Nirenberg inequality. When $s=2N$, the existence and nonexistence of minimizers of $ e_{s}(\rho)$ will also be given. During the arguments, we provide the detailed proof of the $N$-Laplacian Gagliardo-Nirenberg inequality and $N$-Laplacian Pohozaev identity.

Key words: $N$-Laplacian, Gagliardo-Nirenberg inequality, minimizer