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

Abstract

We study the bifurcation of radially symmetric solutions of Δ+ f ( u )=0 on n -balls, into asymmetric ones. We show that if u 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.