Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian

2011/07/12 by Eleonora Cinti, Cinti, Eleonora
Computer Science · Mathematics · #35J20 #35J61 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35J20 #msc:35J61

paper · pdf · doi:10.48550/arxiv.1107.2306

arxiv created 2011/07/12 · openalex publication_date 2011/07/12 · arxiv updated 2011/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation (-Δ)^1/2u=f(u) in all the space \re^2m, where f is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate. More precisely, we prove the existence of a saddle-shaped solution in every even dimension 2m, as well as its monotonicity properties, asymptotic behaviour, and instability in dimensions 2m=4 and 2m=6. These results are relevant in connection with the analog for fractional equations of a conjecture of De Giorgi on the 1-D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1-D solutions, to be global minimizers in high dimensions, a property not yet established.

Citations

• Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations

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