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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Stefano Pigola, Marco Rigoli and Alberto G. Setti
Journal: Trans. Amer. Math. Soc. 362 (2010), 1907-1936
MSC (2000): Primary 58J05, 58J50; Secondary 35J60, 35P05, 53C21
Published electronically: November 13, 2009
MathSciNet review: 2574881
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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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Additional Information

Stefano Pigola
Affiliation: Dipartimento di Fisica e Matematica, Università dell’Insubria - Como, via Valleggio 11, I-22100 Como, Italy
Email: stefano.pigola@uninsubria.it

Marco Rigoli
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italy
Email: rigoli@mat.unimi.it

Alberto G. Setti
Affiliation: Dipartimento di Fisica e Matematica, Università dell’Insubria - Como, via Valleggio 11, I-22100 Como, Italy
Email: alberto.setti@uninsubria.it

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04752-7
PII: S 0002-9947(09)04752-7
Keywords: Nonlinear elliptic equations, existence and non-existence results, eigenvalues, Riemannian manifolds.
Received by editor(s): February 17, 2006
Received by editor(s) in revised form: September 7, 2007
Published electronically: November 13, 2009
Dedicated: Dedicated to the memory of Franca Burrone Rigoli
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.