# Persistence of Coron’s solution in nearly critical problems

• Volume: 6, Issue: 2, page 331-357
• ISSN: 0391-173X

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We consider the problem$\left\{\begin{array}{cc}-\Delta u={u}^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \setminus \epsilon \omega ,\hfill \\ u>0\hfill & \text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\phantom{\rule{4.0pt}{0ex}}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right$/extract_itex]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. ## How to cite top Musso, Monica, and Pistoia, Angela. "Persistence of Coron’s solution in nearly critical problems." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.2 (2007): 331-357. <http://eudml.org/doc/272293>. @article{Musso2007, 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&gt;0 & \text\{in \}\Omega \setminus \varepsilon \omega ,\\ u=0 & \text\{on \} \partial \left( \Omega \setminus \varepsilon \omega \right) ,\end\{array\}\right.\}$where $\Omega$ and $\omega$ are smooth bounded domains in $\mathbb \{R\}^N$, $N\ge 3$, $\varepsilon &gt;0$ 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.},
author = {Musso, Monica, Pistoia, Angela},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {nonlinear elliptic equations; Dirichlet condition; Coron's solution},
language = {eng},
number = {2},
pages = {331-357},
publisher = {Scuola Normale Superiore, Pisa},
title = {Persistence of Coron’s solution in nearly critical problems},
url = {http://eudml.org/doc/272293},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Musso, Monica
AU - Pistoia, Angela
TI - Persistence of Coron’s solution in nearly critical problems
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 2
SP - 331
EP - 357
AB - We consider the problem${\left\lbrace \begin{array}{ll}-\Delta u= u^{{N+2\over N-2}+\lambda } & \text{in }\Omega \setminus \varepsilon \omega , \\ u&gt;0 & \text{in }\Omega \setminus \varepsilon \omega ,\\ u=0 & \text{on } \partial \left( \Omega \setminus \varepsilon \omega \right) ,\end{array}\right.}$where $\Omega$ and $\omega$ are smooth bounded domains in $\mathbb {R}^N$, $N\ge 3$, $\varepsilon &gt;0$ 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.
LA - eng
KW - nonlinear elliptic equations; Dirichlet condition; Coron's solution
UR - http://eudml.org/doc/272293
ER -

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