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, p. 623-664

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation (-Δ) 1/2 u=f(u) in the whole space 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.

Published online : 2019-02-21
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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