June  2005, 4(2): 341-356. doi: 10.3934/cpaa.2005.4.341

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

Received  June 2004 Revised  November 2004 Published  March 2005

We consider a class of equations of the form

$ -\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.

Citation: Andrés Ávila, Louis Jeanjean. A result on singularly perturbed elliptic problems. Communications on Pure & Applied Analysis, 2005, 4 (2) : 341-356. doi: 10.3934/cpaa.2005.4.341
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