Persistence of Coron’s solution in nearly critical problems

Monica Musso; Angela Pistoia

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

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

Abstract

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We consider the problem - Δ u = u N + 2 N - 2 + λ in Ω ∖ ε ω , u > 0 in Ω ∖ ε ω , u = 0 on ∂ Ω ∖ ε ω , where Ω and ω are smooth bounded domains in ℝ N , N ≥ 3 , ε > 0 and λ ∈ ℝ . We prove that if the size of the hole ε goes to zero and if, simultaneously, the parameter λ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

How to cite

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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 -

References

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