On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energy

Abstract

We study the radial symmetry of minimizers to the Schrödinger–Poisson–Slater (S–P–S) energy: \inf\limits_{\scriptsize \begin{array}{l} u \in H^{1}(\mathbb{R}^{3}) \\ \|u\|_{L^{2}(\mathbb{R}^{3})} = \rho \end{array}} \frac{1}{2}\int \limits_{\mathbb{R}^{3}}|\mathrm{∇}u|^{2} + \frac{1}{4}\int \limits_{\mathbb{R}^{3}}\int \limits_{\mathbb{R}^{3}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x−y|}\:dx\:dy−\frac{1}{p}\int \limits_{\mathbb{R}^{3}}|u|^{p}\:dx provided that 2 < p < 3 and ρ is small. The main result shows that minimizers are radially symmetric modulo suitable translation. Résumé On montre la radialité des minimiseurs de lʼénergie de Schrödinger–Poisson–Slater \inf\limits_{\scriptsize \begin{array}{l} u \in H^{1}(\mathbb{R}^{3}) \\ \|u\|_{L^{2}(\mathbb{R}^{3})} = \rho \end{array}} \frac{1}{2}\int \limits_{\mathbb{R}^{3}}|\mathrm{∇}u|^{2} + \frac{1}{4}\int \limits_{\mathbb{R}^{3}}\int \limits_{\mathbb{R}^{3}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x−y|}\:dx\:dy−\frac{1}{p}\int \limits_{\mathbb{R}^{3}}|u|^{p}\:dx pourvu que 2 < p < 3 et ρ est petit.