The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: where is a bounded open domain, , and is the critical Sobolev exponent for the embedding . The uncertainty of the sign of in has some interest in itself. We w...

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Bibliographic Details
Published inAdvanced nonlinear studies Vol. 23; no. 1; pp. 6133 - 6162
Main Authors Deng, Yinbin, He, Qihan, Pan, Yiqing, Zhong, Xuexiu
Format Journal Article
LanguageEnglish
Published De Gruyter 21.02.2023
Subjects
35A01
35A15
35B09
35B33
35G30
Brézis-Nirenberg problem
critical exponents
logarithmic perturbation
positive solution
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ISSN2169-0375
2169-0375
DOI10.1515/ans-2022-0049

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Summary:We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: where is a bounded open domain, , and is the critical Sobolev exponent for the embedding . The uncertainty of the sign of in has some interest in itself. We will show the existence of positive ground state solution, which is of mountain pass type provided and . While the case of is thornier. However, for , , we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for and if . Comparing with the results in the study by Brézis and Nirenberg ( , Comm. Pure Appl. Math. (1983), 437–477), some new interesting phenomenon occurs when the parameter on logarithmic perturbation is not zero.
ISSN:2169-0375
2169-0375
DOI:10.1515/ans-2022-0049