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Symmetry results for positive solutions of some elliptic equations on manifolds. (English) Zbl 0968.58015
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.

MSC:
58J05 Elliptic equations on manifolds, general theory
35J60 Nonlinear elliptic equations
35B50 Maximum principles in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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