Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-28T13:38:58.645Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Afrouzi, G.A. and Brown, K.J., ‘Unbounded principal eigenfunctions for problems on all RN’, Differential Integral Equations 14 (2001), 37–50.CrossRefGoogle Scholar
[2]Afrouzi, G.A. and Brown, K.J., ‘On a diffusive logistic equation’, J. Math. Anal. Appl. 225 (1998), 326–339.CrossRefGoogle Scholar
[3]Bandle, C. and Marcus, M., ‘Asymptotic behavior of solutions and their derivatives for semilinear elliptic problems with blow-up on the boundary’, Ann. Inst. H. Poincaré Anal. Nonlinéaire 12 (1995), 155–171.CrossRefGoogle Scholar
[4]Berestycki, H., Nirenberg, L. and Varadhan, S.R.S., ‘The principal engenvalue and maximum principle for second-order elliptic operator in general domains’, Comm. Pure. Appl. Math. 47 (1994), 47–92.CrossRefGoogle Scholar
[5]Bianchi, B. and Rigoli, M., ‘Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds’, Trans. Amer. Math. Soc. 349 (1997), 4753–4774.CrossRefGoogle Scholar
[6]Brown, K. J., Cosner, C. and Fleckinger, J., ‘Principal eigenvalues for problems with indefinite function on RN’, Proc. Amer. Math. Soc. 109 (1990), 147–155.Google Scholar
[7]Du, Y. and Li, S., ’Positive solutions with prescribed patterns in some simple semilinear euqations’, Differential Integral Equations 15 (2002), 805–822.CrossRefGoogle Scholar
[8]Du, Y. and Ma, L., ‘Logistic type equations on RN by a squeezing method involving boundary blow-up solutions’, J. London, Math. Soc. (2) 64 (2001), 107–124.CrossRefGoogle Scholar
[9]Du, Y. and Ma, L., ‘Positive solutions of an elliptic partial differential equation on RN’, J. Math. Anal. Appl. 271 (2002), 409–425.CrossRefGoogle Scholar
[10]Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order, (second edition) (Springer-Verlag, Berlin, 1983).Google Scholar
[11]Marcus, M. and Veron, L., ‘Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equation’, Ann. Inst. H. Poincaré Anal. Nonlinéaire 14 (1997), 237–274.CrossRefGoogle Scholar
[12]Ratto, A., Rigoli, M. and Veron, L., ‘Scalar curvature and conformal deformations of noncompact Riemannian manifolds’, Math. Z. 225 (1997), 395–426.CrossRefGoogle Scholar