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

Zhi-Yun Tang; Xianhua Tang; Zhi-Yun Tang; Xianhua Tang

doi:10.3934/cam.2025019

Communications in Analysis and Mechanics

2025, Volume 17, Issue 2: 462-473. doi: 10.3934/cam.2025019

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Research article

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

Received: 18 May 2024 Revised: 08 February 2025 Accepted: 15 April 2025 Published: 07 May 2025
35D99, 35J15, 35J91

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.
- Hardy-Sobolev critical exponents,
- positive solutions,
- mountain pass lemma,
- the least action principle,
- method of sub-supersolutions
Citation: Zhi-Yun Tang, Xianhua Tang. Positive solutions for critical singular elliptic equations without Ambrosetti-Rabinowitz type conditions[J]. Communications in Analysis and Mechanics, 2025, 17(2): 462-473. doi: 10.3934/cam.2025019

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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.

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