Geodesic
Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb{R}^{2m}$
Journal of the European Mathematical Society, Tome 11 (2009) no. 4, pp. 819-843.

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We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation −∆ u = f(u) in the whole R2_m_, where f is of bistable type. It is known that in dimension 2_m_ = 2 there exists a saddle-shaped solution. This is a solution which changes sign in R2 and vanishes only on {|x_1 | = |x_2 |}. 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 2_m = 4. More precisely, our main result establishes that if 2_m = 4, every solution vanishing on the Simons cone {(_x_1, _x_2) ∈ R2 × R2 : |_x_1| = |_x_2|} 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.
DOI : 10.4171/jems/168
Classification : 35-XX, 53-XX, 00-XX
Keywords: Allen–Cahn equation, saddle-shaped solutions, Simons cone, instability, Morse index, conjecture of De Giorgi on 1D symmetry 
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     author = {Xavier Cabr\'e and Joana Terra},
     title = {Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb{R}^{2m}$},
     journal = {Journal of the European Mathematical Society},
     pages = {819--843},
     publisher = {mathdoc},
     volume = {11},
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     year = {2009},
     doi = {10.4171/jems/168},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/168/}
}
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Xavier Cabré; Joana Terra. Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb{R}^{2m}$. Journal of the European Mathematical Society, Tome 11 (2009) no. 4, pp. 819-843. doi : 10.4171/jems/168. http://geodesic.mathdoc.fr/articles/10.4171/jems/168/

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