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Abstract:

We consider the elliptic equation -Δu = f(u) in the whole ℝ2m, where f is of bistable type. It is known that there exists a saddle-shaped solution in ℝ2m. This is a solution which changes sign in ℝ2m and vanishes only on the Simons cone C = {(x1, x2) ε ℝm ×ℝm: {pipe}x1{pipe} = {pipe}x2{pipe}} It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution. These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established. © Taylor & Francis Group, LLC.

Registro:

Documento: Artículo
Título:Qualitative properties of saddle-shaped solutions to bistable diffusion equations
Autor:Cabré, X.; Terra, J.
Filiación:Departament de Matemàtica Aplicada I, ICREA and Universitat Politècnica de Catalunya, Barcelona, Spain
Departamento de Matemática, Universidad de Buenos Aires, Ciudad Universitaria, Buenos Aires, Argentina
Palabras clave:Asymptotic behavior; Bistable elliptic diffusion equations; Monotonicity properties; Saddle-shaped solutions; Stability of solutions
Año:2010
Volumen:35
Número:11
Página de inicio:1923
Página de fin:1957
DOI: http://dx.doi.org/10.1080/03605302.2010.484039
Título revista:Communications in Partial Differential Equations
Título revista abreviado:Commun. Partial Differ. Equ.
ISSN:03605302
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03605302_v35_n11_p1923_Cabre

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Citas:

---------- APA ----------
Cabré, X. & Terra, J. (2010) . Qualitative properties of saddle-shaped solutions to bistable diffusion equations. Communications in Partial Differential Equations, 35(11), 1923-1957.
http://dx.doi.org/10.1080/03605302.2010.484039
---------- CHICAGO ----------
Cabré, X., Terra, J. "Qualitative properties of saddle-shaped solutions to bistable diffusion equations" . Communications in Partial Differential Equations 35, no. 11 (2010) : 1923-1957.
http://dx.doi.org/10.1080/03605302.2010.484039
---------- MLA ----------
Cabré, X., Terra, J. "Qualitative properties of saddle-shaped solutions to bistable diffusion equations" . Communications in Partial Differential Equations, vol. 35, no. 11, 2010, pp. 1923-1957.
http://dx.doi.org/10.1080/03605302.2010.484039
---------- VANCOUVER ----------
Cabré, X., Terra, J. Qualitative properties of saddle-shaped solutions to bistable diffusion equations. Commun. Partial Differ. Equ. 2010;35(11):1923-1957.
http://dx.doi.org/10.1080/03605302.2010.484039