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 −∆ u = f(u) in the whole ℝ2_m_, where f is of bistable type. It is known that in dimension 2_m_ = 2 there exists a saddle-shaped solution. This is a solution which changes sign in ℝ2 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 2_m = 4. More precisely, our main result establishes that if 2_m = 4, every solution vanishing on the Simons cone {(_x_1, _x_2) ∈ ℝ2 × ℝ2 : |_x_1| = |_x_2|} 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.