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

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.