Abstract

We consider the problem $${\left\lbrace \begin{array}{ll}-\Delta u= u^{{N+2\over N-2}+\lambda } & \text{in }\Omega \setminus \varepsilon \omega, \\ u>0 & \text{in }\Omega \setminus \varepsilon \omega,\\ u=0 & \text{on } \partial \left,\end{array}\right.}$$ where $\Omega $ and $\omega $ are smooth bounded domains in $\mathbb{R}^N$, $N\ge 3$, $\varepsilon >0$ and $\lambda \in \mathbb{R}.$ We prove that if the size of the hole $\varepsilon $ goes to zero and if, simultaneously, the parameter $\lambda $ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin