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.
Pigola, S., Rigoli, M., Setti, A. (2010). Existence and non-existence results for a logistic-type equation on manifolds. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 362(4), 1907-1936 [10.1090/S0002-9947-09-04752-7].
Existence and non-existence results for a logistic-type equation on manifolds
Pigola S.;
2010
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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