The nonlinear Schrödinger equation in the half-space

Antonio J. Fernández; Tobias Weth

doi:10.1007/s00208-020-02129-8

The nonlinear Schrödinger equation in the half-space

Antonio J. Fernández, Tobias Weth

Department of Mathematical Sciences

Johann Wolfgang Goethe-Universitat Frankfurt am Main

Research output: Contribution to journal › Article › peer-review

5 Citations (SciVal)

Abstract

The present paper is concerned with the half-space Dirichlet problem [Equation not available: see fulltext.]where R+N:={x∈RN:xN>0} for some N≥ 1 and p> 1 , c> 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (P_c). We prove that the existence and multiplicity of bounded positive solutions to (P_c) depend in a striking way on the value of c> 0 and also on the dimension N. We find an explicit number cp∈(1,e), depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions N≥ 2 , we prove that, for 0 < c< c_p, problem (P_c) admits infinitely many bounded positive solutions, whereas, for c> c_p, there are no bounded positive solutions to (P_c).

Original language	English
Pages (from-to)	361-397
Number of pages	37
Journal	Mathematische Annalen
Volume	383
Issue number	1-2
Early online date	16 Jan 2021
DOIs	https://doi.org/10.1007/s00208-020-02129-8
Publication status	Published - 30 Jun 2022

Bibliographical note

Funding Information:
The authors wish to thank the anonymous referees for their valuable comments and corrections. Part of this work was done while the first author was visiting the Goethe-Universität Frankfurt. He wishes to thank his hosts for the warm hospitality and the financial support.

Publisher Copyright:
© 2021, The Author(s).

Funding

The authors wish to thank the anonymous referees for their valuable comments and corrections. Part of this work was done while the first author was visiting the Goethe-Universität Frankfurt. He wishes to thank his hosts for the warm hospitality and the financial support.

ASJC Scopus subject areas

General Mathematics

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10.1007/s00208-020-02129-8Licence: CC BY

Cite this

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title = "The nonlinear Schr{\"o}dinger equation in the half-space",

abstract = "The present paper is concerned with the half-space Dirichlet problem [Equation not available: see fulltext.]where R+N:={x∈RN:xN>0} for some N≥ 1 and p> 1 , c> 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (Pc). We prove that the existence and multiplicity of bounded positive solutions to (Pc) depend in a striking way on the value of c> 0 and also on the dimension N. We find an explicit number cp∈(1,e), depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions N≥ 2 , we prove that, for 0 < c< cp, problem (Pc) admits infinitely many bounded positive solutions, whereas, for c> cp, there are no bounded positive solutions to (Pc).",

author = "Fern{\'a}ndez, {Antonio J.} and Tobias Weth",

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AU - Fernández, Antonio J.

AU - Weth, Tobias

N1 - Funding Information: The authors wish to thank the anonymous referees for their valuable comments and corrections. Part of this work was done while the first author was visiting the Goethe-Universität Frankfurt. He wishes to thank his hosts for the warm hospitality and the financial support. Publisher Copyright: © 2021, The Author(s).

PY - 2022/6/30

Y1 - 2022/6/30

N2 - The present paper is concerned with the half-space Dirichlet problem [Equation not available: see fulltext.]where R+N:={x∈RN:xN>0} for some N≥ 1 and p> 1 , c> 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (Pc). We prove that the existence and multiplicity of bounded positive solutions to (Pc) depend in a striking way on the value of c> 0 and also on the dimension N. We find an explicit number cp∈(1,e), depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions N≥ 2 , we prove that, for 0 < c< cp, problem (Pc) admits infinitely many bounded positive solutions, whereas, for c> cp, there are no bounded positive solutions to (Pc).

AB - The present paper is concerned with the half-space Dirichlet problem [Equation not available: see fulltext.]where R+N:={x∈RN:xN>0} for some N≥ 1 and p> 1 , c> 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (Pc). We prove that the existence and multiplicity of bounded positive solutions to (Pc) depend in a striking way on the value of c> 0 and also on the dimension N. We find an explicit number cp∈(1,e), depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions N≥ 2 , we prove that, for 0 < c< cp, problem (Pc) admits infinitely many bounded positive solutions, whereas, for c> cp, there are no bounded positive solutions to (Pc).

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DO - 10.1007/s00208-020-02129-8

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JO - Mathematische Annalen

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IS - 1-2

ER -

The nonlinear Schrödinger equation in the half-space

Abstract

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ASJC Scopus subject areas

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