Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions
Extending their results on symmetry-breaking for positive solutions for semilinear elliptic equations under Dirichlet conditions [Arch. Ration. Mech. Anal. 95, 217-225 (1986)], the authors study now the problem \(\Delta u+f(u)=0\) on balls \(B_ R\subset {\mathbb{R}}^ n\) with general boundary conditions \(\alpha u+\beta (\partial u/\partial u)=0\) on \(\partial B_ R\). The symmetry is said to break if one has local bifurcation (parameter R) from radially symmetric solutions. Especially for homogeneous Neumann conditions, they give a theorem that under certain technical assumptions on f there is an infinite sequence of radial solutions, at which the symmetry breaks and the dimension of the asymmetric manifold is at least n. In fact the kernels of the corresponding linearized operators have increasing dimensions, showing that the situation is more complicated than in the Dirichlet case. For the general case, bounds for the dimensions of the asymmetric manifolds and theorems on the structure of the kernels of the linearized operators are given.
scientific article; zbMATH DE number 3983695