Nous considérons l'équation de Schrödinger non-linéaire associée à un potentiel singulier de la forme , avec , sur un domaine éventuellement non borné. Nous employons des méthodes d'énergie appropriées pour montrer que si et si les données (initiale et source) ont un support compact alors toute solution doit également avoir un support compact pour tout . Cette propriété contraste avec le comportement des solutions associées aux potentiels réguliers . Des résultats similaires sont également établis pour le problème stationnaire associé et pour les solutions auto-similaires sur l'espace entier et le potentiel . L'existence des solutions est obtenue par des méthodes de compacité sous certaines conditions.
We consider the nonlinear Schrödinger equation associated to a singular potential of the form , for some , on a possible unbounded domain. We use some suitable energy methods to prove that if and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any . This property contrasts with the behavior of solutions associated to regular potentials . Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential . The existence of solutions is obtained by some compactness methods under additional conditions.
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@article{CRMATH_2006__342_7_459_0, author = {B\'egout, Pascal and D{\'\i}az, Jes\'us Ildefonso}, title = {On a nonlinear {Schr\"odinger} equation with a localizing effect}, journal = {Comptes Rendus. Math\'ematique}, pages = {459--463}, publisher = {Elsevier}, volume = {342}, number = {7}, year = {2006}, doi = {10.1016/j.crma.2006.01.027}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2006.01.027/} }
TY - JOUR AU - Bégout, Pascal AU - Díaz, Jesús Ildefonso TI - On a nonlinear Schrödinger equation with a localizing effect JO - Comptes Rendus. Mathématique PY - 2006 SP - 459 EP - 463 VL - 342 IS - 7 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2006.01.027/ DO - 10.1016/j.crma.2006.01.027 LA - en ID - CRMATH_2006__342_7_459_0 ER -
%0 Journal Article %A Bégout, Pascal %A Díaz, Jesús Ildefonso %T On a nonlinear Schrödinger equation with a localizing effect %J Comptes Rendus. Mathématique %D 2006 %P 459-463 %V 342 %N 7 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2006.01.027/ %R 10.1016/j.crma.2006.01.027 %G en %F CRMATH_2006__342_7_459_0
Bégout, Pascal; Díaz, Jesús Ildefonso. On a nonlinear Schrödinger equation with a localizing effect. Comptes Rendus. Mathématique, Tome 342 (2006) no. 7, pp. 459-463. doi : 10.1016/j.crma.2006.01.027. http://archive.numdam.org/articles/10.1016/j.crma.2006.01.027/
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