On a nonlinear Schrödinger equation with a localizing effect

Pascal Bégout and Jesus Ildefonso Diaz
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Pascal Bégout: LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité
Jesus Ildefonso Diaz: UCM - Universidad Complutense de Madrid = Complutense University of Madrid [Madrid]

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Abstract: We consider the nonlinear Schr¨odinger equation associated to a singular potential of the form a|u|−(1−m)u + bu, for some m ∈ (0, 1), on a possible unbounded domain. We use some suitable energy methods to prove that if Re(a) + Im(a) > 0 and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any t > 0. This property contrasts with the behavior of solutions associated to regular potentials (m > 1). Related results are proved also for the associated stationary problem and for self- similar solution on the whole space and potential a|u|−(1−m)u. The existence of solutions is obtained by some compactness methods under additional conditions. To cite this article: Pascal B´egout, Jes´us Ildefonso D´ıaz, C. R. Acad. Sci. Paris, S´er. I 342 (2006).

Date: 2006-03-06
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Published in Comptes Rendus. Mathématique, 2006, 342 (7), pp.459-463. ⟨10.1016/j.crma.2006.01.027⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-05495426

DOI: 10.1016/j.crma.2006.01.027

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