Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations |
| |
Authors: | X Cabré J Terra |
| |
Institution: | 1. Departament de Matemàtica Aplicada I , ICREA and Universitat Politècnica de Catalunya , Barcelona, Spain xavier.cabre@upc.edu;3. Departamento de Matemática , Universidad de Buenos Aires , Ciudad Universitaria, Buenos Aires, Argentina |
| |
Abstract: | We consider the elliptic equation ? Δu = f(u) in the whole ?2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?2m . This is a solution which changes sign in ?2m and vanishes only on the Simons cone 𝒞 = {(x 1, x 2) ∈ ? m × ? 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. |
| |
Keywords: | Asymptotic behavior Bistable elliptic diffusion equations Monotonicity properties Saddle-shaped solutions Stability of solutions |
|
|