Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb{R}^{2m}$
| dc.creator | Cabre, Xavier | |
| dc.creator | Terra, Joana | |
| dc.date | 2008-01-22 | |
| dc.date.accessioned | 2026-07-07T08:55:50Z | |
| dc.date.available | 2026-07-07T08:55:50Z | |
| dc.description | We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation $-Δu = f(u)$ in the whole $\R^{2m}$, where $f$ is of bistable type. It is known that in dimension $2m=2$ there exists a saddle-shaped solution. This is a solution which changes sign in $\R^2$ and vanishes only on $\{|x_1|=|x_2|\}$. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension $2m=4$. More precisely, our main result establishes that if $2m=4$, every solution vanishing on the Simons cone $\{(x^1,x^2)\in\R^m\times\R^m : |x^1|=|x^2|\}$ is unstable outside of every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem. | |
| dc.identifier | https://arxiv.org/abs/0801.3379 | |
| dc.identifier | http://arxiv.org/abs/0801.3379 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/146401 | |
| dc.subject | Analysis of PDEs | |
| dc.title | Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb{R}^{2m}$ | |
| dc.type | text |