Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations.
Merle, Frank (1998)
Documenta Mathematica
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Merle, Frank (1998)
Documenta Mathematica
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Antonio Ambrosetti (2008)
Bollettino dell'Unione Matematica Italiana
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We survey some recent results dealing with some classes of systems of nonlinear Schrödinger equations.
Zhengping Wang, Huan-Song Zhou (2009)
Journal of the European Mathematical Society
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Nakao Hayashi, Masayoshi Tsutsumi (1981)
Mathematische Zeitschrift
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Eugenio Montefusco, Benedetta Pellacci, Marco Squassina (2008)
Journal of the European Mathematical Society
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We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.
N. Hayashi, K. Nakamitsu, M. Tsutsumi (1986)
Mathematische Zeitschrift
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Filip Ficek (2023)
Archivum Mathematicum
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Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials. ...
Patrick Gérard, Vittoria Pierfelice (2010)
Bulletin de la Société Mathématique de France
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We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness...
Nakao Hayashi (1986)
Manuscripta mathematica
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Tohru Ozawa, Jian Zhai (2008)
Annales de l'I.H.P. Analyse non linéaire
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Mejjaoli, Hatem (2011)
Serdica Mathematical Journal
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2000 Mathematics Subject Classification: Primary 42A38. Secondary 42B10. The purpose of this paper is to study the dispersive properties of the solutions of the Dunkl-Schrödinger equation and their perturbations with potential. Furthermore, we consider a few applications of these results to the corresponding nonlinear Cauchy problems.