Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications (Q2884644)

The authors study the symmetry of entire solutions for nonlinear equations involving general operators \(L u(x) = f(u(x))\) in \(\mathbb{R}^n\), with \( \displaystyle \lim_{x_n \to \pm \infty} u(x', x_n) = \pm 1\) uniformly in \( x' \in \mathbb{R}^{n-1}.\) They prove a famous conjecture by Gibbons (also related to a conjecture of De Giorgi) for nonlocal operators \(L\) (e.g. the fractional Laplacian), as well as for nonlinear operators.