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

TL;DR

This paper investigates the bifurcation and symmetry breaking of solutions in nonlinear elliptic PDEs related to functional inequalities, providing asymptotic analysis, local expansions, and numerical insights into global solution behaviors.

Contribution

It introduces new analytical and numerical methods to study symmetry breaking and bifurcation structures in nonlinear elliptic equations of Caffarelli-Kohn-Nirenberg type.

Findings

Identification of bifurcation points and asymptotic behavior of solution branches.

Local expansion characterizing non-symmetric solutions near bifurcation points.

Numerical scenarios illustrating global symmetry breaking phenomena.

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 behavior of the branches for large values of the bifurcation parameter. We also perform an expansion in a neighborhood of the first bifurcation point on the branch of symmetric solutions, that characterizes the local behavior 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 a new light on the symmetry breaking phenomenon.