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Comptes Rendus Mathématique
Volume 341, n° 4
pages 223-228 (août 2005)
Doi : 10.1016/j.crma.2005.06.026
Received : 5 June 2005 ;  accepted : 6 June 2005
Nonlinear Schrödinger equations: concentration on weighted geodesics in the semi-classical limit
Equations de Schrödinger non linéaires : Concentration sur des géodésiques pondérées dans la limite semi-classique
 

Manuel del Pino a , Michał Kowalczyk b, a , Juncheng Wei c
a Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile 
b Kent State University, Department of Mathematical Sciences, Kent, OH 44242, USA 
c Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong 

Abstract

We consider the problem
2u-V(x)u+up=0,u>0,uH1(R2), where  ,   is a small parameter and V is a uniformly positive, smooth potential. Let Γ be a closed curve, nondegenerate geodesic relative to the weighted arclength  , where  . We prove the existence of a solution   concentrating along the whole of Γ , exponentially small in at any positive distance from it, provided that is small and away from certain critical numbers. This proves a conjecture raised in [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys. 235 (2003) 427-466] in the two-dimensional case. To cite this article: M. del Pino et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The full text of this article is available in PDF format.
Résumé

On considère le problème
2u-V(x)u+up=0,u>0,uH1(R2), avec  , où   est un petit paramètre et V est un potentiel régulier, uniformément positif. Soit Γ une courbe fermée formant une géodésique non dégénérée relativement à la longueur pondérée  , avec  . Nous démontrons lʼexistence dʼune solution   qui se concentre le long de la courbe Γ tout entière, exponentiellement petite en à toute distance positive de Γ , pourvu que soit petit et évite certaines valeurs critiques. Ceci répond affirmativement à une conjecture énoncée dans [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys. 235 (2003) 427-466] dans le cas bi-dimensionnel. Pour citer cet article : M. del Pino et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The full text of this article is available in PDF format.


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