Symmetry-breaking for positive solutions of semilinear elliptic equations
Consider the Dirichlet problem \[ (*)\quad \Delta u(x)+f(u(x))=0,\quad x\in B_ R;\quad u(x)=0,\quad x\in \partial B_ R \] where \(f\in C^ 1\) and \(B_ R\subset {\mathbb{R}}^ n\) denotes the open ball of radius \(R>0\). From the results of Gidas, Ni and Nirenberg all positive solutions u of (*) are radial functions such that \(u_ r(r)<0\), \(0<r<R\) and consequently they solve the problem \[ u_{rr}+((n-1)/r)u_ r+f(u)=0,\quad 0<r<R;\quad u_ r(0)=u(R)=0. \] So it is natural to think of solutions of (*) as symmetric functions. In the paper a few results providing necessary and also sufficient conditions for symmetry breaking and infinitesimal symmetry breaking are given. A bifurcation Theorem of Vanderbauwhede which extends an earlier result of Crandall and Rabinowitz is used.
scientific article; zbMATH DE number 4023839