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On the radiality of constrained minimizers to the Schrödinger-Poisson-Slater energy. (English. French summary) Zbl 1260.35204

Summary: We study the radial symmetry of minimizers to the Schrödinger-Poisson-Slater energy: \[ \text{inf}_{\substack{ u\in H^1(\mathbb{R}^3)\\ \| u\|_{L^2(\mathbb{R}^3)}=\rho}}\;{1\over 2} \int_{\mathbb{R}^3} |\nabla u|^2+{1\over 4} \int_{\mathbb{R}^3}\, \int_{\mathbb{R}^3} {|u(x)|^2|u(y)|^2\over |x-y|}\,dx\,dy-{1\over p} \int_{\mathbb{R}^3} |u|^p\,dx \] provided that \(2< p< 3\) and \(\rho\) is small. The main result shows that minimizers are radially symmetric modulo suitable translation.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J20 Variational methods for second-order elliptic equations
35B06 Symmetries, invariants, etc. in context of PDEs
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