## 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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