Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian
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Submission date: 13. Jul. 2011
published in: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 12 (2013) 3, p. 623-665
DOI number (of the published article): 10.2422/2036-2145.201107_002
MSC-Numbers: 35J61, 35J20, 35B40
Keywords and phrases: half-Laplacian, saddle-shaped solutions, stability properties
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We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation (-Δ)1∕2u = f(u) in all the space ℝ2m, 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.