  In this paper, the following critical biharmonic elliptic problem
\begin{eqnarray*} \begin{cases} \Delta^2u= \lambda u+\mu u\ln
u^2+|u|^{2^{**}-2}u, &x\in\Omega,\\ u=\dfrac{\partial u}{\partial \nu}=0,
&x\in\partial\Omega \end{cases} \end{eqnarray*} is considered, where
$\Omega\subset \mathbb{R}^{N}$ is a bounded smooth domain with $N\geq5$. Some
interesting phenomenon occurs due to the uncertainty of the sign of the
logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the
problem admits at lest one nontrivial weak solution under some appropriate
assumptions of $\lambda$ and $\mu$. Moreover, a nonexistence result is also
obtained. Comparing the results in this paper with the known ones, one sees
that some new phenomena occur when the logarithmic perturbation is introduced.

Analysis of PDEs