Existence and non-existence of minimizers for Hardy-Sobolev type inequality with Hardy potentials

Jann Long Chern; Masato Hashizume; Gyeongha Hwang

doi:10.1080/00036811.2023.2268659

Existence and non-existence of minimizers for Hardy-Sobolev type inequality with Hardy potentials

Jann Long Chern, Masato Hashizume, Gyeongha Hwang^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : (Formula presented.) where (Formula presented.), Ω is a smooth domain, (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.). Concerning the coefficients of Hardy potentials, we derive a sharp threshold for the existence and non-existence of a minimizer. In addition, we study the existence and non-existence of a positive solution to the Euler-Lagrangian equations corresponding to the minimization problems.

Original language	English
Pages (from-to)	1831-1845
Number of pages	15
Journal	Applicable Analysis
Volume	103
Issue number	10
DOIs	https://doi.org/10.1080/00036811.2023.2268659
Publication status	Published - 2024

Keywords

Hardy potential
Semilinear elliptic equation
existence
minimizers of Hardy-Sobolev type inequality
non-existence

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1080/00036811.2023.2268659

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title = "Existence and non-existence of minimizers for Hardy-Sobolev type inequality with Hardy potentials",

abstract = "Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : (Formula presented.) where (Formula presented.), Ω is a smooth domain, (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.). Concerning the coefficients of Hardy potentials, we derive a sharp threshold for the existence and non-existence of a minimizer. In addition, we study the existence and non-existence of a positive solution to the Euler-Lagrangian equations corresponding to the minimization problems.",

keywords = "Hardy potential, Semilinear elliptic equation, existence, minimizers of Hardy-Sobolev type inequality, non-existence",

author = "Chern, {Jann Long} and Masato Hashizume and Gyeongha Hwang",

note = "Publisher Copyright: {\textcopyright} 2023 Informa UK Limited, trading as Taylor & Francis Group.",

year = "2024",

doi = "10.1080/00036811.2023.2268659",

language = "English",

volume = "103",

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journal = "Applicable Analysis",

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TY - JOUR

T1 - Existence and non-existence of minimizers for Hardy-Sobolev type inequality with Hardy potentials

AU - Chern, Jann Long

AU - Hashizume, Masato

AU - Hwang, Gyeongha

PY - 2024

Y1 - 2024

N2 - Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : (Formula presented.) where (Formula presented.), Ω is a smooth domain, (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.). Concerning the coefficients of Hardy potentials, we derive a sharp threshold for the existence and non-existence of a minimizer. In addition, we study the existence and non-existence of a positive solution to the Euler-Lagrangian equations corresponding to the minimization problems.

AB - Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : (Formula presented.) where (Formula presented.), Ω is a smooth domain, (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.). Concerning the coefficients of Hardy potentials, we derive a sharp threshold for the existence and non-existence of a minimizer. In addition, we study the existence and non-existence of a positive solution to the Euler-Lagrangian equations corresponding to the minimization problems.

KW - Hardy potential

KW - Semilinear elliptic equation

KW - existence

KW - minimizers of Hardy-Sobolev type inequality

KW - non-existence

UR - http://www.scopus.com/inward/record.url?scp=85173992339&partnerID=8YFLogxK

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U2 - 10.1080/00036811.2023.2268659

DO - 10.1080/00036811.2023.2268659

M3 - Article

AN - SCOPUS:85173992339

SN - 0003-6811

VL - 103

SP - 1831

EP - 1845

JO - Applicable Analysis

JF - Applicable Analysis

IS - 10

ER -

Existence and non-existence of minimizers for Hardy-Sobolev type inequality with Hardy potentials

Abstract

Keywords

ASJC Scopus subject areas

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