Monotonicity for solutions to semilinear problems in epigraphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912223455084544 |
|---|---|
| author | Beuvin, Nicolas Farina, Alberto Sciunzi, Berardino |
| author_facet | Beuvin, Nicolas Farina, Alberto Sciunzi, Berardino |
| contents | We consider positive solutions, possibly unbounded, to the semilinear equation $-Δu=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when $f$ is a (locally or globally) Lipschitz-continuous function satisfying $ f(0) \geq 0$. As an application of our new monotonicity theorems, we prove some classification and/or non-existence results. To prove our results, we first establish some new comparison principles for semilinear problems on general unbounded open sets of $\mathbb{R}^N$, and then we use them to start and to complete a modified version of the moving plane method adapted to the geometry of the epigraph $Ω$. As a by-product of our analysis, we also prove some new results of uniqueness and symmetry for solutions (possibly unbounded and sign-changing) to the homogeneous Dirichlet BVP for the semilinear Poisson equation in fairly general unbounded domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monotonicity for solutions to semilinear problems in epigraphs Beuvin, Nicolas Farina, Alberto Sciunzi, Berardino Analysis of PDEs We consider positive solutions, possibly unbounded, to the semilinear equation $-Δu=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when $f$ is a (locally or globally) Lipschitz-continuous function satisfying $ f(0) \geq 0$. As an application of our new monotonicity theorems, we prove some classification and/or non-existence results. To prove our results, we first establish some new comparison principles for semilinear problems on general unbounded open sets of $\mathbb{R}^N$, and then we use them to start and to complete a modified version of the moving plane method adapted to the geometry of the epigraph $Ω$. As a by-product of our analysis, we also prove some new results of uniqueness and symmetry for solutions (possibly unbounded and sign-changing) to the homogeneous Dirichlet BVP for the semilinear Poisson equation in fairly general unbounded domains. |
| title | Monotonicity for solutions to semilinear problems in epigraphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.04805 |