Qualitative properties of saddle-shaped solutions to bistable diffusion equations

TL;DR

This paper investigates the stability, asymptotic behavior, and existence of saddle-shaped solutions to a bistable elliptic equation in six-dimensional space, providing insights relevant to De Giorgi's conjecture and minimality properties.

Contribution

It establishes the instability of saddle-shaped solutions in six dimensions, describes their asymptotic behavior, and proves the existence of minimal and maximal solutions with monotonicity properties.

Findings

Saddle-shaped solutions are unstable outside compact sets in 6D.

Existence of minimal and maximal saddle-shaped solutions.

Maximal solutions exhibit specific monotonicity properties.

Abstract

We consider the elliptic equation $-\Delta u = f(u)$ in the whole $\R^{2m}$, where $f$ is of bistable type. It is known that there exists a saddle-shaped solution in $\R^{2m}$. This is a solution which changes sign in $\R^{2m}$ and vanishes only on the Simons cone ${\mathcal C}=\{(x^1,x^2)\in\R^m\times\R^m: |x^1|=|x^2|\}$. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when $2m=6$ every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution. These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.