Comptes Rendus
Partial Differential Equations
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]
Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 223-228.

On considère le problème

ɛ2ΔuV(x)u+up=0,u>0,uH1(R2),
avec p>1, où ɛ>0 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 ΓVσ, avec σ=p+1p112. Nous démontrons l'existence d'une solution uε qui se concentre le long de la courbe Γ tout entière, exponentiellement petite en ɛ à toute distance positive de Γ, pourvu que ɛ 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.

We consider the problem

ɛ2ΔuV(x)u+up=0,u>0,uH1(R2),
where p>1, ɛ>0 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 ΓVσ, where σ=p+1p112. We prove the existence of a solution uε concentrating along the whole of Γ, exponentially small in ɛ at any positive distance from it, provided that ɛ 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.06.026

Manuel del Pino 1 ; Michał Kowalczyk 1, 2 ; Juncheng Wei 3

1 Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
2 Kent State University, Department of Mathematical Sciences, Kent, OH 44242, USA
3 Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
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     title = {Nonlinear {Schr\"odinger} equations: concentration on weighted geodesics in the semi-classical limit},
     journal = {Comptes Rendus. Math\'ematique},
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Manuel del Pino; Michał Kowalczyk; Juncheng Wei. Nonlinear Schrödinger equations: concentration on weighted geodesics in the semi-classical limit. Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 223-228. doi : 10.1016/j.crma.2005.06.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.026/

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