A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2019
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916710698713088 |
|---|---|
| author | Filippucci, Roberta Pucci, Patrizia Souplet, Philippe |
| author_facet | Filippucci, Roberta Pucci, Patrizia Souplet, Philippe |
| contents | We consider the elliptic equation $-Δu = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$.
We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant.
The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument.
This solves, in the case of bounded solutions,
a problem left open in~\cite{BVGHV}, where the authors consider the case $0<p<2$. Some extensions to elliptic systems are also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_06816 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient Filippucci, Roberta Pucci, Patrizia Souplet, Philippe Analysis of PDEs We consider the elliptic equation $-Δu = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in~\cite{BVGHV}, where the authors consider the case $0<p<2$. Some extensions to elliptic systems are also given. |
| title | A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1907.06816 |