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The existence of infinitely many solutions all bifurcating from λ = 0

  • Wolfgang Rother (a1)
Synopsis

We consider the non-linear differential equation

and state conditions for the function q such that (*) has infinitely many distinct pairs of (weak) solutions such that holds for all k ∈ ℕ. The main tools are results from critical point theory developed by A. Ambrosetti and P. H. Rabinowitz [1].

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References
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1Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 359381.
2Küpper, T. and Riemer, D.. Necessary and sufficient conditions for bifurcation from the continuous spectrum. Nonlinear anal. 3 (1979), 555561.
3Rother, W.. Bifurcation of nonlinear elliptic equations on ℝN. Bull. London Math. Soc. 21 (1989), 567572.
4Rother, W.. Bifurcation for a semilinear elliptic equation on ℝN with radially symmetric coefficients. Manuscripta math. 65 (1989), 413426.
5Ruppen, H.-J.. The existence of infinitely many bifurcation branches. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 307320.
6Stuart, C. A.. Bifurcation pour des problèmes de Dirichlet et de Neumann sans valeurs propres. C.R. Acad. Sci. Paris. 288 (1979), 761764.
7Stuart, C. A.. Bifurcation for variational problems when the linearisation has no eigenvalues. J. Fund. Anal. 38 (1980), 169187.
8Stuart, C. A.. Bifurcation from the continuous spectrum in the L2-theory of elliptic equations on ℝN. Recent Methods in Nonlinear Analysis and Applications. Proceedings of SAFA IV, Liguori, Napoli (1981).
9Stuart, C. A.. Bifurcation for Dirichlet problems without eigenvalues. Proc. London Math. Soc. (3) 45 (1982), 169192.
10Stuart, C. A.. Bifurcation from the essential spectrum. Lecture Notes in Mathematics 1017 (1983), 575596.
11Stuart, C. A.. A variational approach to bifurcation in L p on an unbounded symmetrical domain. Math. Ann. 263 (1983), 5159.
12Stuart, C. A.. Bifurcation in L P(ℝN) for a semilinear elliptic equation. Proc. London Math. Soc. (3)57 (1988), 511541.
13Toland, J. F.. Global bifurcation for Neumann problems without eigenvalues. J. Differential Equations 44 (1982), 82110.
14Toland, J. F.. Positive solutions of nonlinear elliptic equations – existence and nonexistence of solutions with radially symmetry in L p(∝N). Trans. Amer. Math. Soc. 282 (1984), 335354.
15Zhou, H.-S. and Zhu, X.-P.. Bifurcation from the essential spectrum of superlinear elliptic equations. Appl. Analysis. 28 (1988), 5161.
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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