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Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones. (English) Zbl 1281.35046
In this paper, the construction of entire solutions of the Allen-Cahn equation \[ \Delta u+u - u^{3}=0 \quad \text{in} \;\mathbb{R}^{n+1} \] is considered, where \(n \geq 1\). The existence of bounded stable solutions, whose zero sets are not hyperplanes, is shown for \(n+1\geq 8\).

MSC:
35J61 Semilinear elliptic equations
35B08 Entire solutions to PDEs
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