Symmetry results for positive solutions of some elliptic equations on manifolds (Q1973270)

Adapting the method of moving planes (of A. Alexandrov), the authors study positive solutions of the (nonlinear) Laplace-Beltrami equation \(Lu=-\text{div}(a'|\nabla|^2)\nabla u=f(u)\), on an open submanifold \(\mathcal M\) in an \(n\)-dimensional manifold \(\mathcal N\) in the presence of technical conditions which ensure ellipticity and of some symmetry assumptions. The main regularity assumptions are that \(a\in W^{2,\infty}((0,\infty))\cap{\mathcal C}([0,\infty))\) and that \(f\) is locally Lipschitz. Applications are given for annular domains in \({\mathbb R}^2\), convex geodesic balls in \(S^n\) and in (the hyperbolic space) \(H^n\), subgroups of the polarized Heisenberg group, and for monotonicity results.