Uniqueness and multiplicity for semilinear elliptic problems in unbounded domains
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913704700805120 |
|---|---|
| author | Berestycki, Henri Graham, Cole Wei, Juncheng |
| author_facet | Berestycki, Henri Graham, Cole Wei, Juncheng |
| contents | We study the influence of geometry on semilinear elliptic equations of bistable or nonlinear-field type in unbounded domains. We discover a surprising dichotomy between epigraphs that are bounded from below and those that contain a cone of aperture greater than $π$: the former admit at most one positive bounded solution, while the latter support infinitely many. Nonetheless, we show that every epigraph admits at most one strictly stable solution. To prove uniqueness, we strengthen the method of moving planes by decomposing the domain into one region where solutions are stable and another where they enjoy a form of compactness. Our construction of many solutions exploits a connection with Delaunay surfaces in differential geometry, and extends to all domains containing a suitably wide cone, including exterior domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness and multiplicity for semilinear elliptic problems in unbounded domains Berestycki, Henri Graham, Cole Wei, Juncheng Analysis of PDEs 35J61, 35J25, 35B35, 35B53 (primary), 35B09, 35B50 (secondary) We study the influence of geometry on semilinear elliptic equations of bistable or nonlinear-field type in unbounded domains. We discover a surprising dichotomy between epigraphs that are bounded from below and those that contain a cone of aperture greater than $π$: the former admit at most one positive bounded solution, while the latter support infinitely many. Nonetheless, we show that every epigraph admits at most one strictly stable solution. To prove uniqueness, we strengthen the method of moving planes by decomposing the domain into one region where solutions are stable and another where they enjoy a form of compactness. Our construction of many solutions exploits a connection with Delaunay surfaces in differential geometry, and extends to all domains containing a suitably wide cone, including exterior domains. |
| title | Uniqueness and multiplicity for semilinear elliptic problems in unbounded domains |
| topic | Analysis of PDEs 35J61, 35J25, 35B35, 35B53 (primary), 35B09, 35B50 (secondary) |
| url | https://arxiv.org/abs/2502.16780 |