A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems

TitleA supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems
Publication TypeJournal Article
Year of Publication2020
AuthorsNgô, QAnh
Secondary AuthorsNguyen, VHoang
JournalJournal of Differential Equations
Volume268
Pagination5996-6032
ISSN0022-0396
KeywordsHigher order elliptic problems, Optimizer, Sharp constant, Supercritical Sobolev inequality
Abstract

A Sobolev type embedding for radially symmetric functions on the unit ball B in Rn, n≥3, into the variable exponent Lebesgue space L2⋆+|x|α(B), 2⋆=2n/(n−2), α>0, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequalitysup⁡∫B|u(x)|2⋆+|x|αdx:u∈H0,rad1(B),‖∇u‖L2(B)=1<+∞ holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on B, namely, the embeddingH0,radm(B)↪L2m⋆+|x|α(B) with 2≤m0 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.

URLhttps://www.sciencedirect.com/science/article/pii/S0022039619305480
DOI10.1016/j.jde.2019.11.014