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 $μ,λ,τ\in \mathbb{R}$, $|μ|+|τ|\ne 0$, $Δ^2=ΔΔ$ denotes the iterated N-dimensional Laplacian, $Ω\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $\partial Ω$, $2^{**}=\frac{2N}{N-4}(N\ge5)$ is the critical Sobolev exponent for the embedding $H_{0}^{2}(Ω)\hookrightarrow L^{2^{**}}(Ω)$ and $H_0^2 (Ω)$ is the closure of $C_0^ \infty (Ω)$ under the norm $|| u ||:=(\int_Ω|Δu|^2)^\frac{1}{2}$. 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 $μΔu$, $λu$ and $τu \log u^2$ 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 $λ, μ$ and $τ$.