  In this paper, we consider the existence of nontrivial solutions to the
following critical biharmonic problem with a logarithmic term \begin{equation*}
\begin{cases} \Delta^2 u=\mu \Delta u+\lambda u+|u|^{2^{**}-2}u+\tau u\log u^2,
\ \ x\in\Omega, u|_{\partial \Omega }=\frac{\partial u}{\partial
n}|_{\partial\Omega}=0, \end{cases} \end{equation*} where $\mu,\lambda,\tau \in
\mathbb{R}$, $|\mu|+|\tau|\ne 0$, $\Delta ^2=\Delta \Delta $ denotes the
iterated N-dimensional Laplacian, $\Omega \subset \mathbb{R}^{N}$ is a bounded
domain with smooth boundary $\partial \Omega $, $2^{**}=\frac{2N}{N-4}(N\ge5)$
is the critical Sobolev exponent for the embedding
$H_{0}^{2}(\Omega)\hookrightarrow L^{2^{**}}(\Omega)$ and $H_0^2 (\Omega )$ is
the closure of $C_0^ \infty (\Omega )$ under the norm $|| u
||:=(\int_{\Omega}|\Delta u|^2)^\frac{1}{2}$. The uncertainty of the sign of
$s\log s^2$ in $(0,+\infty)$ has some interest in itself. To know which of the
three terms $\mu \Delta u$, $\lambda u$ and $\tau u \log u^2$ has a greater
influence on the existence of nontrivial weak solutions, we prove the existence
of nontrivial weak solutions to the above problem for $N\ge5$ under some
assumptions of $\lambda, \mu$ and $\tau$.
