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

Jean Dolbeault (1) , Maria J. Esteban (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 branch for large values of the bifurcation parameter. We also perform a formal 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 scenarii. This sheds a new light on the symmetry breaking phenomenon.

Mots clés: Hardy-Sobolev inequality; Caffarelli-Kohn-Nirenberg inequality; extremal functions; Emden-Fowler transformation; radial symmetry; symmetry breaking; ground state; Pöschl-Teller operator; bifurcation; formal expansion; elliptic equations; branches of solutions

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Maria J. Esteban. Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations. 2013. ⟨hal-00812996v1⟩