Communications in Mathematical Physics

, Volume 105, Issue 3, pp 415-441

First online:

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

  • Joel A. SmollerAffiliated withDepartment of Mathematics, University of Michigan
  • , Arthur G. WassermanAffiliated withDepartment of Mathematics, University of Michigan

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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.