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

Publié le :
Classification : 35J61, 35J20, 35B40, 35B08
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     title = {Saddle-shaped solutions of bistable elliptic equations involving the {half-Laplacian}},
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Cinti, Eleonora. Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 623-664. http://www.numdam.org/item/ASNSP_2013_5_12_3_623_0/

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