TY - JOUR

T1 - The nonlinear Schrödinger equation in the half-space

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).

UR - http://www.scopus.com/inward/record.url?scp=85100201187&partnerID=8YFLogxK

U2 - 10.1007/s00208-020-02129-8

DO - 10.1007/s00208-020-02129-8

M3 - Article

AN - SCOPUS:85100201187

VL - 383

SP - 361

EP - 397

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1-2

ER -