Saddle-shaped solutions of bistable diffusion equations in all of ${\mathbb{R}}^{2m}$

Abstract

We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation $-\Delta u=f(u)$ in the whole ${\mathbb{R}}^{2m}$, 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 ${\mathbb{R}}^2$ 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 $\{(x^1,x^2)\in {\mathbb{R}}^m\times {\mathbb{R}}^m:|x^1|=|x^2|\}$ 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.