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Research Article | Open Access

Positive solutions for critical singular elliptic equations without Ambrosetti-Rabinowitz type conditions

Zhi-Yun Tang, Xianhua Tang( )
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, People's Republic of China
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Abstract

We study the subcritical approximations to Li–Lin's open problem, proposed by Li and Lin (Arch Ration Mech Anal 203(3): 943-968, 2012). By applying the variational method, we obtain two positive solutions. We establish a nonexistence theorem for positive solutions. Finally, through the combination of the variational method and the sub-supersolution method, we find a global bifurcation phenomenon for positive solutions.

Keywords

Hardy-Sobolev critical exponentspositive solutionsmountain pass lemmathe least action principlemethod of sub-supersolutions
CLC number: 35D99, 35J15, 35J91

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Communications in Analysis and Mechanics
Volume 17 Issue 2,
June 2025
Pages 462-473
DOI: 10.3934/cam.2025019

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Cite this article:
Tang Z-Y, Tang X. Positive solutions for critical singular elliptic equations without Ambrosetti-Rabinowitz type conditions. Communications in Analysis and Mechanics, 2025, 17(2): 462-473. https://doi.org/10.3934/cam.2025019

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Received: 18 May 2024
Revised: 08 February 2025
Accepted: 15 April 2025
Published: 07 May 2025
©2025 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)