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The Liouville property and a conjecture of De Giorgi. (English) Zbl 1072.35526
The authors study bounded entire solutions of the equation \(\Delta u+u-u^3=0\) in \({\mathbb R}^d\) and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes. The proof uses a Liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator.

35J60 Nonlinear elliptic equations
35C05 Solutions to PDEs in closed form
Full Text: DOI
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