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

TL;DR

This paper proves the existence of solutions for a critical Kirchhoff equation with a logarithmic perturbation in four dimensions, showing that nonlocal terms can enhance the existence range of solutions.

Contribution

It introduces a new analysis of a critical Kirchhoff problem with a logarithmic perturbation, demonstrating the positive impact of nonlocal terms on solution existence.

Findings

Existence of a local minimum solution.

Existence of a least energy solution.

Nonlocal term enlarges parameter ranges for solutions.

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.