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).

ASJC Scopus subject areas

General Mathematics

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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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EP - 397

JO - Mathematische Annalen

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IS - 1-2

ER -

The nonlinear Schrödinger equation in the half-space

Abstract

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