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 ${\left(-\Delta \right)}^{1/2}u=f\left(u\right)$ 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.

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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