论文标题
与对数扰动的关键双向方程的解决方案的存在和不存在
Existence and nonexistence of solutions to a critical biharmonic equation with logarithmic perturbation
论文作者
论文摘要
在本文中，以下关键的Biharmonic椭圆形问题\ begin {eqnarray*} \ begin {cases}Δ^2u =λu=λU=λu+μu\μu\ ln u^2+| U |^| u |^{2^{**} {**} -2} -2} -2 ＆x \ in \partialΩ\ end {cases} \ end {eqnarray*}，其中$ω\ subset \ mathbb {r}^{n} $是一个有界的平滑域，具有$ n \ geq5 $。由于对数项的不确定性，发生了一些有趣的现象。主要是通过使用山通行引理表明，该问题在某些适当的假设$λ$和$μ$的情况下以一种非平凡的弱解决方案接受。此外，还获得了不存在的结果。将本文中的结果与已知的结果进行比较，人们发现当引入对数扰动时发生了一些新现象。
In this paper, the following critical biharmonic elliptic problem \begin{eqnarray*} \begin{cases} Δ^2u= λu+μu\ln u^2+|u|^{2^{**}-2}u, &x\inΩ,\\ u=\dfrac{\partial u}{\partial ν}=0, &x\in\partialΩ\end{cases} \end{eqnarray*} is considered, where $Ω\subset \mathbb{R}^{N}$ is a bounded smooth domain with $N\geq5$. Some interesting phenomenon occurs due to the uncertainty of the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at lest one nontrivial weak solution under some appropriate assumptions of $λ$ and $μ$. Moreover, a nonexistence result is also obtained. Comparing the results in this paper with the known ones, one sees that some new phenomena occur when the logarithmic perturbation is introduced.
