English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We establish existence and qualitative properties of saddle-shaped solutionsof the elliptic fractional equation $(-\Delta)^{1/2}u=f(u)$ in all the space$\re^{2m}$, where $f$ is of bistable type. These solutions are odd with respectto the Simons cone and even with respect to each coordinate. More precisely, weprove the existence of a saddle-shaped solution in every even dimension $2m$,as well as its monotonicity properties, asymptotic behaviour, and instabilityin dimensions $2m=4$ and $2m=6$. These results are relevant in connection withthe analog for fractional equations of a conjecture of De Giorgi on the 1-Dsymmetry of certain solutions. Saddle-shaped solutions are the simplestcandidates, besides 1-D solutions, to be global minimizers in high dimensions,a property not yet established.

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