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Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball

Authors: Alfonso Castro and Alexandra Kurepa
Journal: Proc. Amer. Math. Soc. 101 (1987), 57-64
MSC: Primary 35J65
MathSciNet review: 897070
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Abstract: In this paper we show that a radially symmetric superlinear Dirichlet problem in a ball has infinitely many solutions. This result is obtained even in cases of rapidly growing nonlinearities, that is, when the growth of the nonlinearity surpasses the critical exponent of the Sobolev embedding theorem. Our methods rely on the energy analysis and the phase-plane angle analysis of the solutions for the associated singular ordinary differential equation.

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  • [1] R. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [2] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), 1-32. MR 621969 (82j:35059)
  • [3] A. Bahri and P. L. Lions, Remarques sur la theorie variationnelle des points critiques et applications, C. R. Acad. Sci. Paris 301 (1985), 145-149. MR 801948 (86j:58021)
  • [4] A. Castro and A. C. Lazer, On periodic solutions of weakly coupled systems of differential equations, Boll. Un. Mat. Ital. B (5) 18 (1981), 733-742. MR 641732 (83b:34046)
  • [5] A. Castro and R. Shivaji, Multiple solutions for a Dirichlet problem with jumping nonlinearities. II, J. Math. Anal. Appl. (to appear). MR 954725 (89e:34031)
  • [6] M. J. Esteban, Multiple solutions of semilinear elliptic problems in a ball, J. Differential Equations 57 (1985), 112-137. MR 788425 (87f:35100)
  • [7] S. Fucik and V. Lovicar, Periodic solutions of the equation $ x''(t) + g(x(t)) = p(t)$, Časopis Pěst. Mat. 100 (1975), 160-175. MR 0385239 (52:6104)
  • [8] P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), 753-769. MR 662065 (83k:35037)
  • [9] M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math. 32 (1980), 335-364. MR 595426 (82e:58030)
  • [10] -, Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math. 39 (1982), 233-240. MR 682450 (84a:35097)

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Keywords: Superlinear Dirichlet problem, radially symmetric solution, singular ordinary differential equations, phase-plane analysis, growth condition, rapidly growing nonlinearities
Article copyright: © Copyright 1987 American Mathematical Society

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