A critical elliptic equation with a logarithmic type perturbation
In this note, we consider a critical elliptic equation perturbed by a logarithmic type subcritical term in $\mathbb{R}^4$, and investigate how the logarithmic term affects the existence of weak solutions to such a problem. Since the logarithmic term does not satisfy the standard monotonicity conditi...
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University of Szeged
2025-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11285 |
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| _version_ | 1849318354294145024 |
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| author | Haixia Li Yuzhu Han |
| author_facet | Haixia Li Yuzhu Han |
| author_sort | Haixia Li |
| collection | DOAJ |
| description | In this note, we consider a critical elliptic equation perturbed by a logarithmic type subcritical term in $\mathbb{R}^4$, and investigate how the logarithmic term affects the existence of weak solutions to such a problem. Since the logarithmic term does not satisfy the standard monotonicity condition, essential difficulty arises when one looks for weak solutions to this problem in the variational framework. After some delicate estimates on the logarithmic term we can control the mountain pass level of the corresponding functional so that it satisfies the local compactness condition. Then a positive weak solution follows with the application of the Mountain Pass Lemma and Brézis–Lieb lemma. Our result implies that the logarithmic term plays a positive role for the problem to admit positive solutions. |
| format | Article |
| id | doaj-art-80d10ef7d0fd4d53bfc52134198e3ab9 |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-80d10ef7d0fd4d53bfc52134198e3ab92025-08-20T03:50:53ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752025-01-012025511210.14232/ejqtde.2025.1.511285A critical elliptic equation with a logarithmic type perturbationHaixia Li0https://orcid.org/0000-0003-4583-8395Yuzhu Hanhttps://orcid.org/0000-0003-4215-7586School of Mathematics, Changchun Normal University, Changchun, P.R. ChinaIn this note, we consider a critical elliptic equation perturbed by a logarithmic type subcritical term in $\mathbb{R}^4$, and investigate how the logarithmic term affects the existence of weak solutions to such a problem. Since the logarithmic term does not satisfy the standard monotonicity condition, essential difficulty arises when one looks for weak solutions to this problem in the variational framework. After some delicate estimates on the logarithmic term we can control the mountain pass level of the corresponding functional so that it satisfies the local compactness condition. Then a positive weak solution follows with the application of the Mountain Pass Lemma and Brézis–Lieb lemma. Our result implies that the logarithmic term plays a positive role for the problem to admit positive solutions.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11285criticallogarithmic type perturbationbrézis–lieb lemma |
| spellingShingle | Haixia Li Yuzhu Han A critical elliptic equation with a logarithmic type perturbation Electronic Journal of Qualitative Theory of Differential Equations critical logarithmic type perturbation brézis–lieb lemma |
| title | A critical elliptic equation with a logarithmic type perturbation |
| title_full | A critical elliptic equation with a logarithmic type perturbation |
| title_fullStr | A critical elliptic equation with a logarithmic type perturbation |
| title_full_unstemmed | A critical elliptic equation with a logarithmic type perturbation |
| title_short | A critical elliptic equation with a logarithmic type perturbation |
| title_sort | critical elliptic equation with a logarithmic type perturbation |
| topic | critical logarithmic type perturbation brézis–lieb lemma |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11285 |
| work_keys_str_mv | AT haixiali acriticalellipticequationwithalogarithmictypeperturbation AT yuzhuhan acriticalellipticequationwithalogarithmictypeperturbation AT haixiali criticalellipticequationwithalogarithmictypeperturbation AT yuzhuhan criticalellipticequationwithalogarithmictypeperturbation |