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 ${\mathbb{R}}^{2m}$, 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

@article{ASNSP_2013_5_12_3_623_0, author = {Cinti, Eleonora}, title = {Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 12}, number = {3}, year = {2013}, pages = {623-664}, zbl = {1283.35042}, mrnumber = {3137458}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2013_5_12_3_623_0} }

Cinti, Eleonora. Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 3, pp. 623-664. http://www.numdam.org/item/ASNSP_2013_5_12_3_623_0/

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