Extremal Functions for Trudinger-Moser Type Inequalities in ℝN

Abstract

Let $N\geq 2$, $\alpha_N=N\omega_{N-1}^{1/(N-1)}$, where $\omega_{N-1}$ denotes the area of the unit sphere in $\mathbb{R}^N$. In this note, we prove that for any $0<\alpha

$$\sup_{u\in W^{1,N}(\mathbb{R}^{N}),\|u\|_{W^{1,N}(\mathbb{R}^{N})}\leq 1}\int_{\mathbb{R}^{N}}|u|^\beta\Big(e^{\alpha |u|^{\frac{N}{N-1}}}-\sum_{j=0}^{N-2}\frac{\alpha^{j}}{j!}|u|^{\frac{Nj}{N-1}}\Big){\rm d}x$$

can be attained by some function $u\in W^{1,N}(\mathbb{R}^N)$ with $\|u\|_{W^{1,N}(\mathbb{R}^N)}=1$. Moreover, when $\alpha\geq\alpha_{N}$, the above supremum is infinity.