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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Symmetry breaking for a class of semilinear elliptic problems


Authors: Mythily Ramaswamy and P. N. Srikanth
Journal: Trans. Amer. Math. Soc. 304 (1987), 839-845
MSC: Primary 35J65; Secondary 35B32, 58E07
MathSciNet review: 911098
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Abstract: We study positive solutions of the Dirichlet problem for $ - \Delta u = {u^p} - \lambda $, $ p > 1$, $ \lambda > 0$, on the unit ball $ \Omega $. We show that there exists a positive solution $ ({u_0},\,{\lambda _0})$ of this problem which satisfies in addition $ \partial {u_0}/\partial n = 0$ on $ \partial \Omega $. We prove also that at $ ({u_0},\,{\lambda _0})$, the symmetry breaks, i.e. asymmetric solutions bifurcate from the positive radial solutions.


References [Enhancements On Off] (What's this?)

  • [1] G. Cerami, Symmetry breaking for a class of semilinear elliptic problems, Nonlinear Anal. 10 (1986), 1-14. MR 820654 (87h:35108)
  • [2] 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)
  • [3] M. Ramaswamy, On the global set of solutions to a nonlinear $ ODE$-theoretical and numerical description, J. Differential Equations 65 (1986), 1-48. MR 859471 (87m:34015)
  • [4] J. A. Smoller and A. G. Wasserman, Existence, uniqueness and nondegeneracy of positive solutions of semilinear elliptic equations, Comm. Math. Phys. 95 (1984), 129-159. MR 760329 (86c:35058)
  • [5] -, Symmetry breaking for positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal. 95 (1986), 217-225. MR 853965 (88e:35080b)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0911098-4
PII: S 0002-9947(1987)0911098-4
Article copyright: © Copyright 1987 American Mathematical Society