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

TL;DR

This paper proves the existence and analyzes properties of saddle-shaped solutions to a fractional elliptic equation involving the half-Laplacian, with implications for symmetry conjectures and minimality in high dimensions.

Contribution

It establishes the existence, monotonicity, asymptotic behavior, and instability of saddle-shaped solutions for fractional elliptic equations in all even dimensions.

Findings

Existence of saddle-shaped solutions in all even dimensions.

Proven monotonicity and asymptotic properties of solutions.

Identified instability in dimensions 4 and 6.

Abstract

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation $(-\Delta)^{1/2}u=f(u)$ 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.