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Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature
A result on singularly perturbed elliptic problems
| 1. | Departamento de Ingeniería Matemática, Universidad de La Frontera, Casilla 54-D, Temuco, Chile |
| 2. | Equipe de Mathématiques (UMR CNRS 6623), Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France |
$ -\varepsilon^2\Delta u + V(x)u = f(u), \quad u\in H^1(\mathbf R^N).$
For a local minimum $x_0$ of the potential $V(x)$, we show that there exists a sequence $\varepsilon_n\to 0$, for which corresponding solutions $u_n(x) \in H^1(\mathbf R^N) $ concentrate at $x_0$. Our assumptions on $f(\xi)$ are mainly the ones under which the associated autonomous problem
$ -\Delta v + V(x_0)v = f(v), \quad v\in H^1(\mathbf R^N),$
admits a non trivial solution.
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