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Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions

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Abstract

We study the bifurcation of radially symmetric solutions of Δ+f(u)=0 onn-balls, into asymmetric ones. We show that ifu satisfies homogeneous Neumann boundary conditions, the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension. For general boundary conditions, we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions, and on the structure of the kernels of the linearized operators.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 48109-1003, Ann Arbor, MI, USA

    Joel A. Smoller & Arthur G. Wasserman

Authors
  1. Joel A. Smoller
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  2. Arthur G. Wasserman
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Communicated by L. Nirenberg

Research supported in part by the NSF under Grant No. MCS-800 2337

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Smoller, J.A., Wasserman, A.G. Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions. Commun.Math. Phys. 105, 415–441 (1986). https://doi.org/10.1007/BF01205935

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  • DOI: https://doi.org/10.1007/BF01205935

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