Existence of nontrivial solutions for critical biharmonic equations with logarithmic term

In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \left\{ \begin{array}{ll} \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{array} \right. \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 on $\lambda, \mu$ and $\tau$.