2009/07/17 by Xavier Cabré, Xavier Cabre, Cabre, Xavier +2
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.0907.3008
arxiv created 2009/07/17 · openalex publication_date 2009/07/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathcal C=\(x1,x2)∈\Rm×\Rm: |x1|=|x2|\. 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.