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arXiv · 2410.09351

Existence of nontrivial solutions to a critical Kirchhoff equation with a logarithmic type perturbation in dimension four

Qian Zhang·Yuzhu Han
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Abstract

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, Br\'{e}zis-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.

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Qian Zhang, Yuzhu Han. 2024-10-12. Existence of nontrivial solutions to a critical Kirchhoff equation with a logarithmic type perturbation in dimension four. https://arxiv.org/abs/2410.09351

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