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Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ≥ 3

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Abstract

The authors prove that flat ground state solutions (i.e. minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions N = 1,2 and they can be stable for N ≥ 3 for suitable values of the involved exponents.

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Authors and Affiliations

  1. Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Jesús Ildefonso Díaz

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Cantoblanco, Madrid, Spain

    Jesús Hernández

  3. Institute of Mathematics, Ufa Science Center of RAS 112, Chernyshevsky Str., Ufa, 450077, Russia

    Yavdat Il’yasov

Authors
  1. Jesús Ildefonso Díaz
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  2. Jesús Hernández
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  3. Yavdat Il’yasov
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Correspondence to Jesús Ildefonso Díaz.

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Dedicated to a master, Haïm Brezis, with admiration

This work was supported by the projects of the DGISPI (Spain) (Ref. MTM2011-26119, MTM2014- 57113) and the UCM Research Group MOMAT (Ref. 910480).

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Díaz, J.I., Hernández, J. & Il’yasov, Y. Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ≥ 3. Chin. Ann. Math. Ser. B 38, 345–378 (2017). https://doi.org/10.1007/s11401-016-1073-2

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  • DOI: https://doi.org/10.1007/s11401-016-1073-2

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