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

Publication Type | Journal Article |

Year of Publication | 2020 |

Authors | Ngô, QAnh |

Secondary Authors | Nguyen, VHoang |

Journal | Journal of Differential Equations |

Volume | 268 |

Pagination | 5996-6032 |

ISSN | 0022-0396 |

Keywords | Higher 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. |

URL | https://www.sciencedirect.com/science/article/pii/S0022039619305480 |

DOI | 10.1016/j.jde.2019.11.014 |