A critical Kirchhoff problem with a logarithmic type perturbation in high dimension

In this paper, the following critical Kirchhoff-type elliptic equation involving a logarithmic-type perturbation

$$-(a+b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{\Delta }u=\lambda |u{|}^{q-2}u\ln |u{|}^2+\mu |u{|}^{2}u$$

is considered in a bounded domain in ${\mathbb{R}}^4$. One of the main obstructions one encounters when looking for weak solutions to Kirchhoff problems in high dimensions is that the boundedness of the $(PS)$ sequence is hard to obtain. By combining a result by Jeanjean ^[27] with the mountain pass lemma and Brézis–Lieb's lemma, it is proved that either the norm of the sequence of approximation solutions goes to infinity or the problem admits a nontrivial weak solution, under some appropriate assumptions on $a$, $b$, $\lambda$, and $\mu$.