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 ���1u D f .u/ in the whole R2m, where f is of bistable type. It is known that in dimension 2m D 2 there exists a saddle-shaped solution. This is a solution which changes sign in R2 and vanishes only on fjx1j D jx2jg. 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 D 4. More precisely, our main result establishes that if 2m D 4, every solution vanishing on the Simons cone f.x1; x2/ 2 Rm Rm : jx1j D jx2jg is unstable outside 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.