Nonlinear Schrödinger equations: concentration on weighted geodesics in the semi-classical limit

del Pino, Manuel; Kowalczyk, Michał; Wei, Juncheng

doi:10.1016/j.crma.2005.06.026

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]

Présenté par : Haïm Brezis

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

Comptes Rendus. Mathématique, Tome 341 (2005) no. 4, pp. 223-228.

Résumé
Abstract

On considère le problème

ɛ^{2} Δ u - V (x) u + u^{p} = 0, u > 0, u \in H^{1} (R^{2}),

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

\int_{Γ} V^{σ}

, avec

σ = \frac{p + 1}{p - 1} - \frac{1}{2}

. Nous démontrons l'existence d'une solution

u_{ε}

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

ɛ^{2} Δ u - V (x) u + u^{p} = 0, u > 0, u \in H^{1} (R^{2}),

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

\int_{Γ} V^{σ}

, where

σ = \frac{p + 1}{p - 1} - \frac{1}{2}

. We prove the existence of a solution

u_{ε}

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 : 2005-06-05
Accepté le : 2005-06-06
Publié le : 2005-08-03

DOI : 10.1016/j.crma.2005.06.026

Affiliations des auteurs :

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

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     title = {Nonlinear {Schr\"odinger} equations: concentration on weighted geodesics in the semi-classical limit},
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del Pino, Manuel; Kowalczyk, Michał; Wei, Juncheng. Nonlinear Schrödinger equations: concentration on weighted geodesics in the semi-classical limit. Comptes Rendus. Mathématique, Tome 341 (2005) no. 4, pp. 223-228. doi : 10.1016/j.crma.2005.06.026. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.026/

Bibliographie
Cité par

[1] Ambrosetti, A.; Badiale, M.; Cingolani, S. Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., Volume 140 (1997), pp. 285-300

[2] Ambrosetti, A.; Malchiodi, A.; Ni, W.-M. Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Comm. Math. Phys., Volume 235 (2003), pp. 427-466

[3] Badiale, M.; D'Aprile, T. Concentration around a sphere for a singularly perturbed Schrödinger equation, Nonlinear Anal., Volume 49 (2002), pp. 947-985

[4] Benci, V.; D'Aprile, T. The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Differential Equations, Volume 184 (2002), pp. 109-138

[5] Byeon, J.; Wang, Z.-Q. Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., Volume 65 (2002), pp. 295-316

[6] del Pino, M.; Felmer, P. Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., Volume 149 (1997), pp. 245-265

[7] del Pino, M.; Felmer, P. Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., Volume 324 (2002) no. 1, pp. 1-32

[8] M. del Pino, M. Kowalczyk, J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., in press

[9] Floer, A.; Weinstein, A. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., Volume 69 (1986), pp. 397-408

[10] Grossi, M. On the number of single-peak solutions of the nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Lineaire, Volume 19 (2002), pp. 261-280

[11] Jeanjean, L.; Tanaka, K. Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, Volume 21 (2004) no. 3, pp. 287-318

[12] Kang, X.; Wei, J. On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, Volume 5 (2000) no. 7–9, pp. 899-928

[13] Levitan, B.M.; Sargsjan, I.S. Sturm–Liouville and Dirac operators, Math. Appl. (Soviet Ser.), vol. 59, Kluwer Academic, Dordrecht, 1991

[14] Malchiodi, A.; Montenegro, M. Boundary concentration phenomena for a singularly perturbed elliptic problem, Commun. Pure Appl. Math., Volume 55 (2002), pp. 1507-1568

[15] Malchiodi, A.; Montenegro, M. Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., Volume 124 (2004) no. 1, pp. 105-143

[16] Malchiodi, A. Solutions concentrating at curves for some singularly perturbed elliptic problems, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 10, pp. 775-780

[17] Oh, Y.-G. On positive multibump states of nonlinear Schrödinger equation under multiple well potentials, Commun. Math. Phys., Volume 131 (1990), pp. 223-253

[18] Wang, X. On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., Volume 153 (1993), pp. 229-243

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