Saddle-shaped Solutions of Bistable Diffusion Equations in All of R

We study the existence and instability properties of saddleshaped solutions of the semilinear elliptic equation −∆u = f(u) in the whole R, 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 and vanishes only on {|x1| = |x2|}. 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, x) ∈ R×R : |x| = |x|} 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.