Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-13T00:36:30.237Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on asymptotic uniqueness for some nonlinearities which change sign

Published online by Cambridge University Press:  17 April 2009

E. N. Dancer
E. N. Dancer
Affiliation:
School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
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.

In this paper, we prove the uniqueness of the decaying positive solution on all of Rn for certain second order non linear elliptic equations. This improves earlier work of a number of authors. These problems occur in the theory of peak solutions. In particular, our results apply to a number of non-smooth nonlinearities which occur as limiting equations in population problems.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 2 , April 2000 , pp. 305 - 312
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Conway, E., Gardner, R. and Smoller, J., ‘Stability and bifurcation of steady state solutions for predator-prey equations’, Adv. in Appl. Math 3 (1982), 288–334.CrossRefGoogle Scholar
[2]Dancer, E.N., ‘On positive solutions of some pairs of equations’, Trans. Amer. Math. Soc 284 (1984), 729–743.CrossRefGoogle Scholar
[3]Dancer, E.N., ‘On the structure of solutions of an equation in catalysis theory when a parameter is large’, J. Differential Equations 37 (1980), 404–437.CrossRefGoogle Scholar
[4]Dancer, E.N., ‘On positive solutions of some pairs of equations II’, J. Differential Equations 60 (1985), 236–258.CrossRefGoogle Scholar
[5]Dancer, E.N., ‘On uniqueness and stability for solutions of singularly perturbed predator-prey type equations with diffusion’, J. Differential Equations 102 (1993), 1–32.CrossRefGoogle Scholar
[6]Dancer, E.N., ‘On positive solutions of some singularly perturbed problems where the nonlinearity changes sign’, Topol. Methods Nonlinear Anal 5 (1995), 141–175.CrossRefGoogle Scholar
[7]Dancer, E.N. and Schmitt, K., ‘On positive solutions of semilinear elliptic problems’, Proc. Amer. Math. Soc 101 (1987), 445–452.CrossRefGoogle Scholar
[8]Du, Y., ‘Bifurcation from semitrivial solution bundles and application to certain equation systems’, Nonlinear Anal 27 (1996), 1407–1435.CrossRefGoogle Scholar
[9]Kaper, H. and Kwong, M.H., ‘Uniqueness of non-negative solutions of a class of semilinear elliptic equations’, in Nonlinear diffusion equations and their equilibrium states II, (Ni, W-M, Peletier, L.A. and Serrin, J., Editors) (Springer-Verlag, Berlin, Heidelberg, New York, 1988), pp. 1–17.Google Scholar
[10]Kwong, M.H. and Zhang, L., ‘Uniqueness of the positive solution of Δu + f (u) = 0 in an annulus’, Differential Integral Equations 4 (1991), 583–599.CrossRefGoogle Scholar
[11]Peletier, L.A. and Serrin, J., ‘Uniqueness of non-negative solutions of semilinear equations in Rn’, J. Differential Equations 61 (1986), 380–397.CrossRefGoogle Scholar