Persistence of Coron's solution in nearly critical problems
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 2, p. 331-357

We consider the problem $\left\{\begin{array}{cc}-\Delta u={u}^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\phantom{\rule{4pt}{0ex}}\Omega \setminus \epsilon \omega ,\hfill \\ u>0\hfill & \text{in}\phantom{\rule{4pt}{0ex}}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\phantom{\rule{4pt}{0ex}}\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.

Classification:  35J60,  35J25
@article{ASNSP_2007_5_6_2_331_0,
author = {Musso, Monica and Pistoia, Angela},
title = {Persistence of Coron's solution in nearly critical problems},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {2},
year = {2007},
pages = {331-357},
zbl = {1147.35041},
mrnumber = {2352522},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_2_331_0}
}

Musso, Monica; Pistoia, Angela. Persistence of Coron's solution in nearly critical problems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 2, pp. 331-357. http://www.numdam.org/item/ASNSP_2007_5_6_2_331_0/

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