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Title:	Saddle-shaped solutions of bistable diffusion equations in all of R2m
Author:	Cabré Vilagut, Xavier; Mourao Terra, Joana
Other authors:	Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
Abstract:	We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation ð€€€1u D f .u/ in the whole R2m, where f is of bistable type. It is known that in dimension 2m D 2 there exists a saddle-shaped solution. This is a solution which changes sign in R2 and vanishes only on fjx1j D jx2jg. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension 2m D 4. More precisely, our main result establishes that if 2m D 4, every solution vanishing on the Simons cone f.x1; x2/ 2 Rm Rm : jx1j D jx2jg is unstable outside every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem.
Abstract:	Peer Reviewed
Subject(s):	-Àrees temàtiques de la UPC::Matemàtiques i estadística -Allen–Cahn equation -saddle-shaped solutions -Simons cone -instability -Morse index -conjecture of De Giorgi on 1D symmetry -Equacions diferencials parcials -Equacions el·líptiques no lineals
Rights:	Attribution-NonCommercial-NoDerivs 3.0 Spain http://creativecommons.org/licenses/by-nc-nd/3.0/es/
Document type:	Article - Published version Article
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