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Zbl 0933.35083
Li, Yanyan; Nirenberg, Louis
The Dirichlet problem for singularly perturbed elliptic equations.
(English)
[J] Commun. Pure Appl. Math. 51, No.11-12, 1445-1490 (1998). ISSN 0010-3640

This remarkable paper is devoted to the Dirichlet problem for a singularly perturbed elliptic equation $$-\varepsilon^2 \Delta\widetilde u+ \widetilde u=\widetilde u^q,\ \widetilde u>0,$$ in a bounded domain $\Omega \subset \bbfR^n$, $\widetilde u|_{\partial \Omega}=0$, where $1<q <\infty$ if $n\in \{1,2\}$ and $1<q< (n+2)/(n-2)$ if $n\ge 3$, $\varepsilon>0$ is a small real parameter. The authors present two main results concerning the existence of a family of solutions $\widetilde u_\varepsilon$ of the problem under consideration. The first result is the following. Given the inequality $\max_{Q \in \partial V}d(Q,\partial \Omega)<\max_{Q\in \partial\overline V}d(Q, \partial \Omega)$, where $d(Q,\partial \Omega)\equiv \text{dist} (Q,\partial \Omega)$, $V$ is an open set and $\overline V\subset\Omega$. Then there exists $\overline \varepsilon>0$ and $\widetilde u_\varepsilon$ for $0<\varepsilon < \overline \varepsilon$ such that $\widetilde u_\varepsilon$ has a unique local maximum point $\widetilde Q_\varepsilon\in V$, $d(\widetilde Q_\varepsilon, \partial \Omega) \to\max_{Q\in \partial\overline V}d(Q, \partial \Omega)$ as $\varepsilon \to 0$ and $\widetilde Q_\varepsilon$ is the unique critical point of $\widetilde u_\varepsilon$ provided that $n\in\{1,2\}$ or $\Omega$ is convex. The second result consists in the following statement. If $V$ is open in $\Omega$, $\overline V\subset\Omega$, $\partial V\subset{\cal O}$ $({\cal O} \subset \Omega)$ and the Brouwer degree $\deg(\nabla d(Q,\partial \Omega), V,0) \ne 0$, then there exists $\overline\varepsilon>0$ and $\widetilde u_\varepsilon$ for $0<\varepsilon <\overline\varepsilon$ such that $\widetilde u_\varepsilon$ has a unique local maximum point $\widetilde Q_\varepsilon\in V$, $d(\widetilde Q_\varepsilon,S)\to 0$ $(S=\Omega \setminus {\cal O})$ as $\varepsilon\to 0$ and also $\widetilde Q_\varepsilon$ is the unique critical point of $\widetilde u_\varepsilon$ provided that $n\in\{1,2\}$ or $\Omega$ is convex.
[Dimitar Kolev (Sofia)]
MSC 2000:
*35J70 Elliptic equations of degenerate type
35B25 Singular perturbations (PDE)
35B38 Critical points

Keywords: Dirichlet problem; singularly perturbed elliptic equations; Brouwer degree; local maximum point; unique critical point

Cited in: Zbl 0966.35039

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