Saddle-shaped solutions of bistable diffusion equations in all of &Rdbl;2m

Abstract

We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation Δu = f(u) in the whole &Rdbl;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 &Rdbl;2 and vanishes only on f{|x1| = |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 {(x1, x2) ε &Rdbl;m × &Rdbl;m : |x1| = |x2|} 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. © European Mathematical Society 2009.