## 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 | |

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

- Mathematics(all)