超临界Bahri-Coron问题的变号爆破解
Sign-changing blowing-up solutions for supercritical Bahri-Coron's problem
摘要：
设$Ω$是$\R^n$，$n\ge 3$中的有界区域，具有光滑边界$\partialΩ$和一个小孔.我们给出了非线性椭圆问题$$ -Δu=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } Ω, \quad \quad u=0 \mbox{ on } \partial Ω, $$的变号{ \it冒泡}解的第一个例子，其中$\ve$是一个小的正参数。构造中的基本单元是临界Yamabe问题$$ -Δw = |w|^{\frac{4}{n-2}} w, \ \ w \in {\mathcal D}^{1,2} (\R^n) $$的变号节点解，该问题具有大量（$3n$）核。
英文摘要：
Let $Ω$ be a bounded domain in $\R^n$, $n\ge 3$ with smooth boundary $\partialΩ$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -Δu=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } Ω, \quad \quad u=0 \mbox{ on } \partial Ω, $$ where $\ve$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$ -Δw = |w|^{\frac{4}{n-2}} w, \ \ w \in {\mathcal D}^{1,2} (\R^n) $$ which has large number ($3n$) of kernels.
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