Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2020-10-02 , DOI: 10.1007/s00041-020-09789-9 Van Hoang Nguyen
Let \(\mathbf{B} \) denote the unit ball in \({\mathbb {R}}^n\) with \(n\ge 2\). In this paper, we present the balance conditions on the nonlinearity function F and the weight function h such that the weighted Trudinger–Moser type inequalities
$$\begin{aligned} \sup _{u \in W^{1,n}_{0}(\mathbf{B }),\, u \text { is radial}, \Vert \nabla u\Vert _{L^n(\mathbf{B })} \le 1} \int _{\mathbf{B }} F(u) h(|x|) dx < \infty \end{aligned}$$holds. We also study the attainability of these inequalities. These results generalizes the ones obtained by De Figueiredo et al. [15] to the higher dimension \(n\ge 3\) as well as weaken the conditions on F and h given in [15]. Our results also extend the ones of Yang and Zhu [29] to more general cases of the nonlinearity function F and the weight function h.
中文翻译：
单位球中权重消失的Trudinger-Moser型不等式
令\（\ mathbf {B} \ ）用\（n \ ge 2 \）表示\（{\ mathbb {R}} ^ n \）中的单位球。在本文中，我们给出了非线性函数F和权重函数h的平衡条件，使得加权的Trudinger-Moser型不等式
$$ \ begin {aligned} \ sup _ {u \ in W ^ {1，n} _ {0}（\ mathbf {B}），\，u \ text {isradial}，\ Vert \ nabla u \ Vert _ {L ^ n（\ mathbf {B}）} \ le 1} \ int _ {\ mathbf {B}} F（u）h（| x |）dx <\ infty \ end {aligned} $$持有。我们还研究了这些不等式的可及性。这些结果概括了De Figueiredo等人获得的结果。[15]到高维\（n \ ge 3 \）以及削弱[15]中给出的F和h条件。我们的结果还将Yang和Zhu [29]的研究扩展到非线性函数F和权重函数h的更一般情况。
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