Persistence of Coron's solution in nearly critical problems

Musso, Monica; Pistoia, Angela

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 2, pp. 331-357.

We consider the problem

$$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u>0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$

where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon >0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ 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.

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