A note on a critical bi-harmonic equation with logarithmic perturbation

In this short note, a critical bi-harmonic elliptic equation with logarithmic perturbation is considered. The same problem has been investigated recently by Li et al. (2023), where existence and nonexistence of weak solutions were obtained when the parameters and the domain satisfy some appropriate conditions. Part of their existence condition is weakened in this note. Moreover, the types of the weak solutions can be specified with the help of Brézis–Lieb’s lemma and Ekeland’s variational principle.