Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations

Abstract

In this paper, we study the bifurcation of branches of non-symmetric solutions from the symmetric branch of solutions to the Euler–Lagrange equations satisfied by optimal functions in functional inequalities of Caffarelli–Kohn–Nirenberg type. We establish the asymptotic behaviour of the branches for large values of the bifurcation parameter. We also perform an expansion in a neighbourhood of the first bifurcation point on the branch of symmetric solutions that characterizes the local behaviour of the non-symmetric branch. These results are compatible with earlier numerical and theoretical observations. Further numerical results allow us to distinguish two global scenarios. This sheds new light on the symmetry breaking phenomenon.