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- W3147968596 abstract "We study the existence and instability properties of saddle-shaped solutions of the semi- linear elliptic equation 1uD f.u/ in the whole R 2m , where f is of bistable type. It is known that in dimension 2mD 2 there exists a saddle-shaped solution. This is a solution which changes sign in R 2 and vanishes only onfjx1jDjx2jg. 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 2mD 4. More precisely, our main result establishes that if 2mD 4, every solution vanishing on the Simons conef.x 1 ;x 2 /2 R m R m :jx 1 jDjx 2 jg 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." @default.
- W3147968596 created "2021-04-13" @default.
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- W3147968596 date "2009-01-01" @default.
- W3147968596 modified "2023-09-26" @default.
- W3147968596 title "Saddle-shaped solutions of bistable diffusion equations in all of R 2m" @default.
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