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Topological Methods in Nonlinear Analysis

On the number of stable positive solutions of weakly nonlinear elliptic equations when the diffusion is small
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On the number of stable positive solutions of weakly nonlinear elliptic equations when the diffusion is small

Authors

  • Edward N. Dancer

DOI:

https://doi.org/10.12775/TMNA.2020.025

Keywords

Small diffusion, stability of solutions to elliptic equations, blow-up

Abstract

We study the exact number of stable positive solutions of weakly nonlinear elliptic equations with small diffusion under rather general conditions on the nonlinearity.

References

G. Alberti, L. Ambrosio and X. Cabre, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), no. 1, 9–33.

E.N. Dancer, On the number of positive solutions of weakly non-linear elliptic equations when a parameter is large, Proc. London Math. Soc. 3 (1986), no. 3, 429–452.

E.N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations 74 (1988), 120–156.

E.N. Dancer, Some notes on the method of moving planes, Bull. Aust. Math. Soc. 46 (1992), no. 03, 425–434.

E.N. Dancer, Stable and finite Morse index solutions on Rn or on bounded domains with small diffusion II, Indiana Univ. Math. J. 53 (2004), 97–408.

E.N. Dancer, Stable and finite Morse index solutions on Rn or on bounded domains with small diffusion, Trans. Amer. Math. Soc. 357 (2005), no. 3, 1225–1243.

E.N. Dancer, Finite Morse index solutions of supercritical problems, J. Reine Angew. Math. 2008 (2008), no. 620, 213–233.

E.N. Dancer, Some remarks on half space problems, Discrete Contin. Dyn. Syst. 25 (2009), 83–88.

E.N. Dancer, New results for finite Morse index solutions on Rn and applications, Adv. Nonlinear Stud. 10 (2010), 581–595.

L. Dupaigne, Stable solutions of elliptic partial differential equations, Monographs and Surveys in Pure and Applied Mathematics, CRC Press, 2011.

G. Sweers, On the maximum of solutions for a semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A Math. 108 (1988), no. 3–4, 357–370.

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Published

2021-05-25

How to Cite

1.
DANCER, Edward N. On the number of stable positive solutions of weakly nonlinear elliptic equations when the diffusion is small. Topological Methods in Nonlinear Analysis. Online. 25 May 2021. Vol. 59, no. 2A, pp. 467 - 474. [Accessed 29 June 2025]. DOI 10.12775/TMNA.2020.025.
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Issue

Vol 59, No 2A (June 2022)

Section

Articles

License

Copyright (c) 2021 Topological Methods in Nonlinear Analysis

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This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

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