Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials

Moroz, Vitaly; Van Schaftingen, Jean

doi:10.1016/j.crma.2009.05.009

Partial Differential Equations

Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials
[Existence et concentration pour l'équation de Schrödinger non linéaire avec des potentiels à décroissance rapide]

Présenté par : Haïm Brezis

Moroz, Vitaly ¹ ; Van Schaftingen, Jean ²

¹ Swansea University, Department of Mathematics, Singleton Park, Swansea SA2 8PP, Wales, United Kingdom
² Université Catholique de Louvain, département de mathématique, chemin du cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

Comptes Rendus. Mathématique, Tome 347 (2009) no. 15-16, pp. 921-926.

Résumé
Abstract

Nous prouvons l'existence de solutions positives non triviales de

- ε^{2} Δ u + V u = u^{p} dans R^{N},

pour ε petit, où

N ⩾ 3

, V est potentiel continu positif qui a un minimum local strictement positif et

\frac{N}{N - 2} < p < \frac{N + 2}{N - 2}

. Nous n'imposons aucune restriction sur le taux de décroisance de V. En particulier, nous couvrons le cas où le support de V est compact.

The existence of positive solutions to

- ε^{2} Δ u + V u = u^{p} in R^{N},

is proved for small ε, where

N ⩾ 3

, V is a nonnegative continuous potential which has a positive local minimum, and

\frac{N}{N - 2} < p < \frac{N + 2}{N - 2}

. No restriction is imposed on the rate of decay of V; this includes in particular the case where V is compactly supported.

Reçu le : 2009-04-06
Accepté le : 2009-05-11
Publié le : 2009-06-12

DOI : 10.1016/j.crma.2009.05.009

Affiliations des auteurs :

Moroz, Vitaly ¹ ; Van Schaftingen, Jean ²

@article{CRMATH_2009__347_15-16_921_0,
     author = {Moroz, Vitaly and Van Schaftingen, Jean},
     title = {Existence and concentration for nonlinear {Schr\"odinger} equations with fast decaying potentials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {921--926},
     publisher = {Elsevier},
     volume = {347},
     number = {15-16},
     year = {2009},
     doi = {10.1016/j.crma.2009.05.009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.05.009/}
}

TY  - JOUR
AU  - Moroz, Vitaly
AU  - Van Schaftingen, Jean
TI  - Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 921
EP  - 926
VL  - 347
IS  - 15-16
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2009.05.009/
DO  - 10.1016/j.crma.2009.05.009
LA  - en
ID  - CRMATH_2009__347_15-16_921_0
ER  -

%0 Journal Article
%A Moroz, Vitaly
%A Van Schaftingen, Jean
%T Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials
%J Comptes Rendus. Mathématique
%D 2009
%P 921-926
%V 347
%N 15-16
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.05.009/
%R 10.1016/j.crma.2009.05.009
%G en
%F CRMATH_2009__347_15-16_921_0

Moroz, Vitaly; Van Schaftingen, Jean. Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials. Comptes Rendus. Mathématique, Tome 347 (2009) no. 15-16, pp. 921-926. doi : 10.1016/j.crma.2009.05.009. http://www.numdam.org/articles/10.1016/j.crma.2009.05.009/

Bibliographie
Cité par

[1] Ambrosetti, A.; Felli, V.; Malchiodi, A. Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., Volume 7 (2005), pp. 117-144

[2] Ambrosetti, A.; Malchiodi, A. Concentration phenomena for nonlinear Schrödinger equations: recent results and new perspectives, Perspectives in Nonlinear Partial Differential Equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, RI, 2007, pp. 19-30

[3] Ambrosetti, A.; Malchiodi, A.; Ruiz, D. Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math., Volume 98 (2006), pp. 317-348

[4] Bonheure, D.; Van Schaftingen, J. Nonlinear Schrödinger equations with potentials vanishing at infinity, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 903-908

[5] Bonheure, D.; Van Schaftingen, J. Bound state solutions for a class of nonlinear Schrödinger equations, Rev. Mat. Iberoamericana, Volume 24 (2008), pp. 297-351

[6] del Pino, M.; Felmer, P. Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, Volume 4 (1996), pp. 121-137

[7] 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

[8] Gidas, B.; Spruck, J. Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., Volume 34 (1981), pp. 525-598

[9] V. Moroz, J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials, Calc. Var., | DOI

Cité par Sources :