A bifurcation diagram of solutions to semilinear elliptic equations with general supercritical growth

Miyamoto Yasuhito

Naito Yūki

Journal of Differential Equations

We study the global bifurcation diagram of the positive solutions to the problem {△u ＋ λf(u) = 0 in B, u=0 on ∂B, where B is the unit ball in RN with N >＿ 3. Under general supercritical growth conditions on f(u), we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f(u), and we exhibit that a bifurcation curve for △u ＋λf(u) = 0 has the same qualitative property as a classical Gel'fand problem △u ＋ λeu = 0 for N >＿ 3 except N = 10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques.

Bifurcation diagram

Joseph-Lundgren exponent

Uniqueness

Singular extremal solution

Super-exponential

eng

Elsevier

© 2024. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/

This is not the published version. Please cite only the published version.

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[DOI] https://doi.org/10.1016/j.jde.2024.06.026 isVersionOf

日本学術振興会

Japan Society for the Promotion of Science

[Crossref Funder] https://doi.org/10.13039/501100001691

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日本学術振興会

Japan Society for the Promotion of Science

[Crossref Funder] https://doi.org/10.13039/501100001691

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[Crossref Funder] https://doi.org/10.13039/501100001691

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23K03167