The nonlinear Schrödinger equation in the half-space

Antonio J. Fernández, Tobias Weth

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4 Citations (SciVal)


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
Issue number1-2
Early online date16 Jan 2021
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).

ASJC Scopus subject areas

  • Mathematics(all)


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