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The Dirichlet problem for singularly perturbed elliptic equations. (English) Zbl 0933.35083

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 \(\Omega \subset \mathbb{R}^n\), \(\widetilde u|_{\partial \Omega}=0\), where \(1<q <\infty\) if \(n\in \{1,2\}\) and \(1<q< (n+2)/(n-2)\) if \(n\geq 3\), \(\varepsilon>0\) 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 \(\Omega\), \(\overline V\subset\Omega\), \(\partial V\subset{\mathcal O}\) \(({\mathcal O} \subset \Omega)\) and the Brouwer degree \(\deg(\nabla d(Q,\partial \Omega), V,0) \neq 0\), 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 \(\widetilde Q_\varepsilon\in V\), \(d(\widetilde Q_\varepsilon,S)\to 0\) \((S=\Omega \setminus {\mathcal O})\) 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.

MSC:

35J70 Degenerate elliptic equations
35B25 Singular perturbations in context of PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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