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Unbounded principal eigenfunctions and the logistic equation on RN

Published online by Cambridge University Press:  17 April 2009

Wei Dong  and
Yihong Du
Wei Dong
Affiliation:
School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351, Australia e-mail: wdong@turing.une.edu.au, ydu@turing.une.edu.au
Yihong Du
Affiliation:
School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351, Australia e-mail: wdong@turing.une.edu.au, ydu@turing.une.edu.au
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Abstract

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We consider the logistic equation − Δu = a (x) u − b (x) up on all of RN with possibly unbounded coefficients near infinity. We show that under suitable growth conditions of the coefficients, the behaviour of the positive solutions of the logistic equation can be largely determined. We also show that certain linear eigenvalue problems on all of RN have principal eigenfunctions that become unbounded near infinity at an exponential rate. Using these results, we finally show that the logistic equation has a unique positive solution under suitable growth restrictions for its coefficients.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 67 , Issue 3 , June 2003 , pp. 413 - 427
Copyright
Copyright © Australian Mathematical Society 2003

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