Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-03-07 , DOI: 10.1007/s00526-020-1705-y Quốc Anh Ngô , Van Hoang Nguyen
Given \(\alpha >0\), we establish the following two supercritical Moser–Trudinger inequalities
$$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$and
$$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( \alpha _n |u|^{\frac{n}{n-1} + |x|^\alpha } \big ) dx < +\infty , \end{aligned}$$where \(W^{1,n}_{0,\mathrm{rad}}(B)\) is the usual Sobolev spaces of radially symmetric functions on B in \({\mathbb {R}}^n\) with \(n\ge 2\). Without restricting to the class of functions \(W^{1,n}_{0,\mathrm{rad}}(B)\), we should emphasize that the above inequalities fail in \(W^{1,n}_{0}(B)\). Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.
中文翻译：
超临界Moser-Trudinger不等式和相关的椭圆问题
给定\（\ alpha> 0 \），我们建立了以下两个超临界Moser-Trudinger不等式
$$ \ begin {aligned} \ mathop {\ sup} \ limits _ {u \ in W ^ {1，n} _ {0，\ mathrm {rad}}（B）：\ int _B | \ nabla u | ^ n dx \ le 1} \ int _B \ exp \ big（（\ alpha _n + | x | ^ \ alpha）| u | ^ {\ frac {n} {n-1}} \ big）dx <+ \ infty \ end {aligned} $$和
$$ \ begin {aligned} \ mathop {\ sup} \ limits _ {u \ in W ^ {1，n} _ {0，\ mathrm {rad}}（B）：\ int _B | \ nabla u | ^ n dx \ le 1} \ int _B \ exp \ big（\ alpha _n | u | ^ {\ frac {n} {n-1} + | x | ^ \ alpha} \ big）dx <+ \ infty，\结束{aligned} $$其中\（W ^ {1，N} _ {0，\ mathrm {弧度}}（B）\）是对径向对称函数通常Sobolev空间乙在\（{\ mathbb {R}} ^ N \）与\（nge 2 \）。在不限制函数\（W ^ {1，n} _ {0，\ mathrm {rad}}（B）\）的类别的情况下，我们应该强调上述不等式在\（W ^ {1，n}中失败_ {0}（B）\）。还研究了关于上述不等式的尖锐性以及最优函数的存在性的问题。为了说明这一发现，提出了对球上一类边值问题的应用。这是我们有关超临界状态下功能不平等的一组工作的第二部分。
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