Existence and non-existence results for a logistic-type equation on manifolds

Pigola, Stefano; Rigoli, Marco; Setti, Alberto

doi:10.1090/S0002-9947-09-04752-7

Existence and non-existence results for a logistic-type equation on manifolds
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by Stefano Pigola, Marco Rigoli and Alberto G. Setti PDF

Trans. Amer. Math. Soc. 362 (2010), 1907-1936 Request permission

Abstract:

We study the steady state solutions of a generalized logistic-type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold, which is mostly taken into account by means of conditions on the volume growth of geodesic balls.

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