Abstract A Sobolev type embedding for radially symmetric functions on the unit ball B in R n , n ≥ 3 , into the variable exponent Lebesgue space L 2 ⋆ + | x | α ( B ) , 2 ⋆ = 2 n / ( n − 2 ) , α > 0 , is known due to J.M. do O, B. Ruf, and P. Ubilla, namely, the inequality sup ⁡ { ∫ B | u ( x ) | 2 ⋆ + | x | α d x : u ∈ H 0 , rad 1 ( B ) , ‖ ∇ u ‖ L 2 ( B ) = 1 } + ∞ holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on B, namely, the embedding H 0 , rad m ( B ) ↪ L 2 m ⋆ + | x | α ( B ) with 2 ≤ m n / 2 , 2 m ⁎ = 2 n / ( n − 2 m ) , and α > 0 holds. Questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied. To illustrate the finding, an application to a boundary value problem on balls driven by polyharmonic operators is presented. This is the first in a set of our works concerning functional inequalities in the supercritical regime.

中文翻译：

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高阶 Sobolev 空间中的超临界 Sobolev 型不等式及相关的高阶椭圆问题

摘要 A Sobolev 型嵌入在单位球 B 上的 R n , n ≥ 3 中的径向对称函数到变指数 Lebesgue 空间 L 2 ⋆ + | × | α ( B ) , 2 ⋆ = 2 n / ( n − 2 ) , α > 0 ，由于 JM do O、B. Ruf 和 P. Ubilla 是已知的，即不等式 sup ⁡ { ∫ B | 你 ( x ) | 2 ⋆ + | × | α dx : u ∈ H 0 , rad 1 ( B ) , ‖ ∇ u ‖ L 2 ( B ) = 1 } + ∞ 成立。在这项工作中，我们将上述不等式推广到 B 上径向对称函数的高阶 Sobolev 空间，即嵌入 H 0 , rad m ( B ) ↪ L 2 m ⋆ + | × | α ( B ) 2 ≤ mn / 2 , 2 m ⁎ = 2 n / ( n − 2 m ) ，并且 α > 0 成立。还研究了关于不等式的锐常数的问题，包括最优函数的存在性。为了说明这一发现，提出了对由多谐算子驱动的球的边值问题的应用。这是我们关于超临界体制中功能不平等的一系列工作中的第一篇。