Existence of nontrivial solutions for critical biharmonic equations with logarithmic term

Abstract

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 $\mu ,\lambda ,\tau \in \mathbb{R}$, $|\mu |+|\tau |\ne 0$, ${\Delta }^2=\Delta \Delta$ denotes the iterated N-dimensional Laplacian, $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $2^{\ast \ast }=\frac{2N}{N-4}(N\ge 5)$ is the critical Sobolev exponent for the embedding $H_0^2(\Omega )\hookrightarrow L^{2^{\ast \ast }}(\Omega )$ 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 $N\ge 5$ under some assumptions of $\lambda ,\mu$ and $\tau$.