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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A counterexample to the nodal domain conjecture and a related semilinear equation
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by Chang Shou Lin and Wei-Ming Ni
Proc. Amer. Math. Soc. 102 (1988), 271-277
DOI: https://doi.org/10.1090/S0002-9939-1988-0920985-9
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Abstract:

In this paper we first establish a nonuniqueness result for a semilinear Dirichlet problem of which the nonlinearity is of super-critical growth. We then apply this result to construct a Schrödinger operator on a domain $\Omega$ such that the second eigenfunctions of this operator (with zero Dirichlet boundary data) have their nodal sets completely contained in the interior of the domain $\Omega$.
References
  • Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR 397805, DOI 10.1007/BF02568142
  • R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
    • -S. Lin, On second eigenfunctions of the Laplacian in ${{\mathbf {R}}^2}$, preprint.
  • Wei-Ming Ni and Roger D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u,r)=0$, Comm. Pure Appl. Math. 38 (1985), no. 1, 67–108. MR 768105, DOI 10.1002/cpa.3160380105
    • -M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations. The analomous case, Proc. Accad. Naz. Lincei 77 (1986), 231-257.
  • Lawrence E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 721–729 (English, with German summary). MR 333487, DOI 10.1007/BF01597076
  • Shing Tung Yau, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. MR 645762
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Bibliographic Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 102 (1988), 271-277
  • MSC: Primary 35B05,; Secondary 35J60
  • DOI: https://doi.org/10.1090/S0002-9939-1988-0920985-9
  • MathSciNet review: 920985