Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

Abstract

We consider the elliptic equation − Δu = f(u) in the whole ℝ2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ℝ2m . This is a solution which changes sign in ℝ2m and vanishes only on the Simons cone 𝒞 = {(x 1, x 2) ∈ ℝ m × ℝ m : |x 1| = |x 2|}. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution. These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.