The nonlinear Schrödinger equation in the half-space

Antonio J. Fernández, Tobias Weth

Research output: Contribution to journalArticlepeer-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 (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).

Original languageEnglish
Pages (from-to)361-397
Number of pages37
JournalMathematische Annalen
Volume383
Issue number1-2
Early online date16 Jan 2021
DOIs
Publication statusPublished - 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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