We study the following biharmonic elliptic problem with slightly subcritical non-power nonlinearity:

$$\begin{aligned} \left\{ \begin{array}{lll} \Delta ^2 u =\frac{|u|^{2^*-2}u}{[\ln (e+|u|)]^\varepsilon }\ \ &{}\textrm{in}\ \Omega , \\ u=\Delta u= 0 \ \ &{} \textrm{on}\ \partial \Omega , \end{array} \right. \end{aligned}$$

where \(2^*=\frac{2n}{n-4}\), \(\Omega \) is a bounded smooth domain in \(\mathbb {R}^n\) with \(n\ge 5\), \(\varepsilon \) is a small positive parameter. By finite-dimensional Lyapunov–Schmidt reduction, we construct a single bubble solution, which concentrates at the non-degenerate critical point of the Robin function.

中文翻译：

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略亚临界非幂非线性的双调和椭圆问题

我们研究以下具有轻微亚临界非幂非线性的双调和椭圆问题：

$$\begin{对齐} \left\{ \begin{array}{lll} \Delta ^2 u =\frac{|u|^{2^*-2}u}{[\ln (e+|u|) )]^\varepsilon }\ \ &{}\textrm{in}\ \Omega , \\ u=\Delta u= 0 \ \ &{} \textrm{on}\ \partial \Omega , \end{array} \正确的。\end{对齐}$$

其中\(2^*=\frac{2n}{n-4}\)，\(\Omega \)是\(\mathbb {R}^n\)中的有界平滑域，其中\(n\ge 5 \) , \(\varepsilon \)是一个小的正参数。通过有限维 Lyapunov-Schmidt 约简，我们构造了一个单气泡解，该解集中在 Robin 函数的非简并临界点。