The nonlinear Schrödinger equation in the half-space
- The present paper is concerned with the half-space Dirichlet problem [...] where ℝ𝑁+:={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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| Author: | Antonio J. Fernández, Tobias WethORCiDGND |
|---|---|
| URN: | urn:nbn:de:hebis:30:3-747421 |
| DOI: | https://doi.org/10.1007/s00208-020-02129-8 |
| ISSN: | 1432-1807 |
| Parent Title (English): | Mathematische Annalen |
| Publisher: | Springer |
| Place of publication: | Berlin ; Heidelberg |
| Document Type: | Article |
| Language: | English |
| Date of Publication (online): | 2021/01/16 |
| Date of first Publication: | 2021/01/16 |
| Publishing Institution: | Universitätsbibliothek Johann Christian Senckenberg |
| Release Date: | 2023/10/25 |
| Volume: | 383 |
| Issue: | 1–2 |
| Page Number: | 37 |
| First Page: | 361 |
| Last Page: | 397 |
| Note: | Open Access funding enabled and organized by Projekt DEAL. |
| HeBIS-PPN: | 515127086 |
| Institutes: | Informatik und Mathematik |
| Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Sammlungen: | Universitätspublikationen |
| Licence (German): |  Creative Commons - CC BY - Namensnennung 4.0 International |
