Sign-changing blowing-up solutions for supercritical Bahri-Coron's problem

Let $\mathrm{\Omega }$ be a bounded domain in ${\mathbb{R}}^n$, $n\ge 3$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{\frac{n+2}{n-2}+\text{ve}-1}uin\mathrm{\Omega },u=0on\mathrm{\partial }\mathrm{\Omega },$$ where $\text{ve}$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$-\mathrm{\Delta }w=\mathrm{\mid }w{\mathrm{\mid }}^{\frac{4}{n-2}}w,w\in {\mathcal{D}}^{1,2}({\mathbb{R}}^n)$$ which has large number ($3n$) of kernels.