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*} \begin{cases} Δ^2 u=μΔu+λu+|u|^{2^{**}-2}u+τu\log u^2, \ \ x\inΩ, u|{\partial Ω}=\frac{\partial u}{\partial n}|{\partialΩ}=0, \end{cases} \end{equation*} where $\mu ,\lambda ,\tau \in \mathbb{R}$, $\mathrm{\mid }\mu \mathrm{\mid }+\mathrm{\mid }\tau \mathrm{\mid }\mathrm{\ne }0$, ${\mathrm{\Delta }}^2=\mathrm{\Delta }\mathrm{\Delta }$ denotes the iterated N-dimensional Laplacian, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$, $2^{\ast \ast }=\frac{2N}{N-4}(N\ge 5)$ is the critical Sobolev exponent for the embedding $H_0^2(\mathrm{\Omega })\hookrightarrow L^{2^{\ast \ast }}(\mathrm{\Omega })$ and $H_0^2(\mathrm{\Omega })$ is the closure of $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ under the norm $\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }\mathrm{\mid }:=({\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\Delta }u{\mathrm{\mid }}^2{)}^{\frac{1}{2}}$. The uncertainty of the sign of $s\log s^2$ in $(0,+\mathrm{\infty })$ has some interest in itself. To know which of the three terms $\mu \mathrm{\Delta }u$, $\lambda u$ 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 $N\ge 5$ under some assumptions of $\lambda ,\mu$ and $\tau$.