Symmetry-breaking at positive solutions
Boundary value problems of the type \[ (*)\quad \Delta w(x)+f(w(x),\lambda)=0,\quad x\in \Omega;\quad w(x)=0,\quad x\in \partial \Omega, \] are considered, where \(f: {\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}}\) is a smooth function, \(\Omega =\{x\in {\mathbb{R}}^ n\); \(\| x\| <1\}\), \(n\geq 2\). If \(\lambda_ 0\), \(w_ 0\) satisfy (*) and \(w_ 0(x)>0\) for all \(x\in \Omega\) then \(w_ 0\) is spherically symmetric. It is proved that under certain assumptions there is a symmetry-breaking bifurcation at \(\lambda_ 0\), \(w_ 0\), i.e. any neighborhood of \(\lambda_ 0\), \(w_ 0\) contains solutions which are not spherically symmetric. Moreover, the bifurcating symmetry-breaking solutions are axisymmetric.
scientific article; zbMATH DE number 4023813