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| We consider the following nonlinear elliptic problems: $$ -\Delta u + u = u^p \ \hbox{in}\ \Omega_\varepsilon, \ \ \ u=0 \ \hbox{on}\ \partial \Omega_\varepsilon. $$ Here $p$ is subcritical and $\Omega_\varepsilon$ is a bounded domain with a smooth boundary which is expanding as $\varepsilon\to 0$. We consider a situation where a cylindrical domain ${\bf R}^k\times D$ appears as a limit of $\Omega_\varepsilon$. We show the existence of a family of solution $(u_\varepsilon)_\varepsilon$ which converges to a solution of the limit problem in a cylindrical domain ${\bf R}^k\times D$ after suitable translations.
This talk is based on my joint works with Jaeyoung Byeon (KAIST, Korea). |
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