# Saddle-shaped solutions of bistable diffusion equations in all of ℝ<sup>2<em>m</em></sup>

### Joana Terra

Universitat Politecnica de Catalunya, Barcelona, Spain### Xavier Cabré

ICREA, Barcelona, Spain

## 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.

## Cite this article

Joana Terra, Xavier Cabré, Saddle-shaped solutions of bistable diffusion equations in all of ℝ<sup>2<em>m</em></sup>. J. Eur. Math. Soc. 11 (2009), no. 4, pp. 819–943

DOI 10.4171/JEMS/168