Qualitative properties of saddle-shaped solutions to bistable diffusion equations

We consider the elliptic equation $-\mathrm{\Delta }u=f(u)$ in the whole ${\mathbb{R}}^{2m}$, where $f$ is of bistable type. It is known that there exists a saddle-shaped solution in ${\mathbb{R}}^{2m}$. This is a solution which changes sign in ${\mathbb{R}}^{2m}$ 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.