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Nonexistence of Multi-bubble Solutions for a Higher Order Mean Field Equation on Convex Domains

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Geometric Properties for Parabolic and Elliptic PDE's

Part of the book series: Springer INdAM Series ((SINDAMS,volume 2))

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Abstract

In this note, we are concerned with the blowing-up behavior of solutions to the 2p-th order mean field equation under the Navier boundary condition:

where Ω is a smooth bounded domain in ℝ2p for p∈ℕ. By using a new Pohozaev type identity for the Green function of (−Δ)p under the Navier boundary condition, we show that the set of blow up points for any blowing-up solution sequence must be a singleton on convex domains, under some assumptions on the weight function V.

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Acknowledgements

This work was partly supported by JSPS Grant-in-Aid for Scientific Research (KAKENHI) (B), No. 23340038.

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Authors and Affiliations

  1. Department of Mathematics & OCAMI, Osaka City University, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Futoshi Takahashi

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  1. Futoshi Takahashi
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Correspondence to Futoshi Takahashi .

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Editors and Affiliations

  1. Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, Firenze, 50124, Italy

    Rolando Magnanini

  2. Graduate School of Information Sciences, Division of Mathematics, Tohoku University, Sendai, 980-8579, Japan

    Shigeru Sakaguchi

  3. , Dipartimento di Matematica ed Applicazio, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, Napoli, 80126, Italy

    Angelo Alvino

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Takahashi, F. (2013). Nonexistence of Multi-bubble Solutions for a Higher Order Mean Field Equation on Convex Domains. In: Magnanini, R., Sakaguchi, S., Alvino, A. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2841-8_18

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