Existence of nontrivial solutions to a critical Kirchhoff equation with a logarithmic type perturbation in dimension four
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913811428016128 |
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| author | Zhang, Qian Han, Yuzhu |
| author_facet | Zhang, Qian Han, Yuzhu |
| contents | In this paper, a critical Kirchhoff equation with a logarithmic type subcritical term is considered in a bounded domain in $\mathbb{R}^4$. We view this problem as a critical elliptic equation with a nonlocal perturbation, and investigate how the nonlocal term affects the existence of weak solutions to the problem. By means of Ekeland's variational principle, Brézis-Lieb's lemma and some convergence tricks for nonlocal problems, we show that this problem admits a local minimum solution and a least energy solution under some appropriate assumptions on the parameters. Moreover, under some further assumptions, the local minimum solution is also a least energy solution. Compared with the ones obtained in [3] and [8], our results show that the introduction of the nonlocal term enlarges the ranges of the parameters such that the problem admits weak solutions, which implies that the nonlocal term has a positive effect on the existence of weak solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of nontrivial solutions to a critical Kirchhoff equation with a logarithmic type perturbation in dimension four Zhang, Qian Han, Yuzhu Analysis of PDEs In this paper, a critical Kirchhoff equation with a logarithmic type subcritical term is considered in a bounded domain in $\mathbb{R}^4$. We view this problem as a critical elliptic equation with a nonlocal perturbation, and investigate how the nonlocal term affects the existence of weak solutions to the problem. By means of Ekeland's variational principle, Brézis-Lieb's lemma and some convergence tricks for nonlocal problems, we show that this problem admits a local minimum solution and a least energy solution under some appropriate assumptions on the parameters. Moreover, under some further assumptions, the local minimum solution is also a least energy solution. Compared with the ones obtained in [3] and [8], our results show that the introduction of the nonlocal term enlarges the ranges of the parameters such that the problem admits weak solutions, which implies that the nonlocal term has a positive effect on the existence of weak solutions. |
| title | Existence of nontrivial solutions to a critical Kirchhoff equation with a logarithmic type perturbation in dimension four |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.09351 |