Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations
Esteban, Maria J.; Dolbeault, Jean (2014), Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations, Nonlinearity, 27, 3, p. n°435. http://dx.doi.org/10.1088/0951-7715/27/3/435
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Abstract (EN)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.
Subjects / Keywordsbranches of solutions; elliptic equations; formal expansion; bifurcation; Pöschl-Teller operator; symmetry breaking; radial symmetry; Emden-Fowler transformation; extremal functions; Caffarelli-Kohn-Nirenberg inequality; Hardy-Sobolev inequality
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