Abstract
We consider the normalized solutions of a Schrodinger system which arises naturally from nonlinear optics, the Hartree–Fock theory for Bose–Einstein condensates. And we investigate the partial symmetry of normalized solutions to the system and their symmetry-breaking phenomena. More precisely, when the underlying domain is bounded and radially symmetric, we develop a kind of polarization inequality with weight to show that the first two components of the normalized solutions are foliated Schwarz symmetric with respect to the same point, while the latter two components are foliated Schwarz symmetric with respect to the antipodal point. Furthermore, by analyzing the singularly perturbed limit profiles of these normalized solutions, we prove that they are not radially symmetric at least for large nonlinear coupling constant $$\beta $$ , which seems a new method to prove the symmetry-breaking phenomenons of normalized solutions.
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