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A counterexample to the nodal domain conjecture and a related semilinear equation


Authors: Chang Shou Lin and Wei-Ming Ni
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
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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 $.


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  • [1 S] -Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), 43-55. MR 0397805 (53:1661)
  • [2 R] Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience, New York, 1953. MR 0065391 (16:426a)
  • [3 B] Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. MR 544879 (80h:35043)
  • [4 C] -S. Lin, On second eigenfunctions of the Laplacian in $ {{\mathbf{R}}^2}$, preprint.
  • [5 W] -M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $ \Delta u + f(u,r) = 0$, Comm. Pure. Appl. Math. 38 (1985), 67-108. MR 768105 (86d:35055)
  • [6 W] -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.
  • [7 L] Payne, On two conjectures in the fixed membrane eigenvalue problem, J. Appl. Math. Phys. (ZAMP) 24 (1973), 720-729. MR 0333487 (48:11812)
  • [8 S] -T. Yau, Problem section, Seminar on Differential Geometry (S.-T. Yau, ed.), Ann. of Math. Studies, no. 102, Princeton Univ. Press, Princeton, N. J., 1982, pp. 669-706. MR 645762 (83e:53029)

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DOI: https://doi.org/10.1090/S0002-9939-1988-0920985-9
Article copyright: © Copyright 1988 American Mathematical Society

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