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Symmetry breaking for a class of semilinear elliptic problems and the bifurcation diagram for a 1-dimensional problem

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Abstract

We study the behaviour of the positive solutions to the Dirichlet problem IRn in the unit ball in IRR wherep<(N+2)/(N−2) ifN≥3 and λ varies over IR. For a special class of functionsg viz.,g(x)=u p0 (x) whereu 0 is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric positive solutions. WhenN=1, we obtain the complete bifurcation diagram for the positive solution curve.

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References

  1. Crandall M. G., Rabinowitz P. H.,Bifurcation from simple eigenvalues, J. Funtctional Analysis8 (1971), 321–340.

    Article  MATH  MathSciNet  Google Scholar 

  2. Crandall M. G., Rabinowitz P. H.,Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal.58 (1975), 207–218.

    Article  MATH  MathSciNet  Google Scholar 

  3. De Figueredo D. G., Lions P. L., Nussbaum R. D.,Apriori estimates and existence of positive solutions of semilinear equations. J. Math. Pures. Appl.61 (1982), 41–63.

    MathSciNet  Google Scholar 

  4. Keller H. B.,Numerical solution of bifurcation and nonlinear eigenvalue problems. “Applications in Bifurcation theory” (P. H. Rabinowitz ed.) Academic Press, New York/London 1977.

    Google Scholar 

  5. McLeod K., Serrin J.,Uniqueness of positive radial solutions of Δu+f(u)=0 in IRn Arch. Rat. Mech. Anal.99, n. 2 (1987), 115–146.

    MATH  MathSciNet  Google Scholar 

  6. Smoller J., Wasserman A.,Symmetry breaking for positive solutions of semilinear elliptic equations., Arch. Rat. Mech. Anal95 (1986), 217–225.

    MATH  MathSciNet  Google Scholar 

  7. Gadam S.,Monotonicity of the forcing term and existence of positive solutions for a class of semilinear elliptic problems. Proc. of AMS, Vol. 113, N. 2, October 1991, 415–418.

    Article  MATH  Google Scholar 

  8. Rabinowitz P. H.,Minimax methods in critical point theory with applications to differential equations, N.65, Regional Conference Series in Mathematics, published by AMS.

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Author notes

  1. Sudhasree Gadam

    Present address: Dept. of Mathematics, Univ. of North Texas, 76203-5116, Denton, TX

Authors and Affiliations

  1. School of Mathematics, Univ. of Minnesota, 127 Vincent Hall, 206 Church st. S.E., 55455, Minneapolis, MN, USA

    Sudhasree Gadam

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  1. Sudhasree Gadam
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Gadam, S. Symmetry breaking for a class of semilinear elliptic problems and the bifurcation diagram for a 1-dimensional problem. Rend. Circ. Mat. Palermo 41, 209–220 (1992). https://doi.org/10.1007/BF02844665

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  • DOI: https://doi.org/10.1007/BF02844665

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