Power law convergence and concavity for the Logarithmic Schrödinger equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915921498472448 |
|---|---|
| author | Gallo, Marco Mosconi, Sunra Squassina, Marco |
| author_facet | Gallo, Marco Mosconi, Sunra Squassina, Marco |
| contents | We study concavity properties of positive solutions to the Logarithmic Schrödinger equation $-Δu=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-Δu = σ\, (u^q-u)$ and build, for any $σ>0$ and $q>1$, solutions $u_q$ such that $u_q^{(1-q)/2}$ is convex. By choosing $σ_q=2/(q-1)$ and letting $q \to 1^+$ we eventually construct a solution $u$ of the Logarithmic Schrödinger equation such that $\log u$ is concave. This seems to be one of the few attempts at studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Power law convergence and concavity for the Logarithmic Schrödinger equation Gallo, Marco Mosconi, Sunra Squassina, Marco Analysis of PDEs 26B25, 35B09, 35B53, 35B99, 35E10, 35J60 We study concavity properties of positive solutions to the Logarithmic Schrödinger equation $-Δu=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-Δu = σ\, (u^q-u)$ and build, for any $σ>0$ and $q>1$, solutions $u_q$ such that $u_q^{(1-q)/2}$ is convex. By choosing $σ_q=2/(q-1)$ and letting $q \to 1^+$ we eventually construct a solution $u$ of the Logarithmic Schrödinger equation such that $\log u$ is concave. This seems to be one of the few attempts at studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks. |
| title | Power law convergence and concavity for the Logarithmic Schrödinger equation |
| topic | Analysis of PDEs 26B25, 35B09, 35B53, 35B99, 35E10, 35J60 |
| url | https://arxiv.org/abs/2411.01614 |