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

In this paper, the following critical Kirchhoff ff-type elliptic equation involving a logarithmic- type perturbation ( Z ) - a + b |Vu|2dx Delta u 2 d x Delta u = lambda |u| q- 2 u ln |u|2 2 + mu |u| 2 u SZ is considered in a bounded domain in R 4 . One of the main obstructions one encounters when looking for weak solutions to Kirchhoff ff problems in high dimensions is that the boundedness of the (PS PS ) sequence is hard to obtain. By combining a result by Jeanjean [27] with the mountain pass lemma and Bre<acute accent>zis-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 .