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

Cinti, Eleonora

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 623-664.

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation ${(-\Delta )}^{1/2}u=f\left(u\right)$ in the whole space ${ℝ}^{2m}$, 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.

Classification : 35J61, 35J20, 35B40, 35B08

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