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Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds. (English) Zbl 1112.58023

The author derives a local gradient estimate for the positive solutions to the equation \[ \Delta u+au\log u+bu=0\quad \text{on}\;M, \] where \(a<0\) and \(b\) are real constants, and \(M\) is a complete non-compact Riemannian manifold. The result obtained is optimal in the case when \((M,g)\) is a complete non-compact expanding gradient Ricci soliton, that is, if \[ Rc=cg+D^2f \] holds with a constant \(c<0,\) where \(Rc\) is the Ricci curvature and \(D^2f\) is the Hessian of the potential \(f\) over \(M.\) It is shown that for a complete non-compact Riemannian manifold \((M,g)\) the local gradient bound of \(f=\log u\) is well controlled by some constants and the lower bound of the Ricci curvature.

MSC:

58J05 Elliptic equations on manifolds, general theory
35B45 A priori estimates in context of PDEs
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