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

Georgiev, Vladimir; Prinari, Francesca; Visciglia, Nicola

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 369-376.

On montre la radialité des minimiseurs de lʼénergie de Schrödinger–Poisson–Slater

$$\underset{\genfrac{}{}{0.0pt}{}{u\in H^1\left({ℝ}^3\right)}{\parallel u{\parallel }_{L^2\left({ℝ}^3\right)}=\rho }}{\inf }\frac{1}{2}\underset{{ℝ}^3}{\int }{\left|\nabla u\right|}^2+\frac{1}{4}\underset{{ℝ}^3}{\int }\underset{{ℝ}^3}{\int }\frac{{\left|u\right(x\left)\right|}^2{\left|u\right(y\left)\right|}^2}{|x-y|}dxdy-\frac{1}{p}\underset{{ℝ}^3}{\int }{\left|u\right|}^{p}dx$$

pourvu que $2<p<3$ et ρ est petit.

We study the radial symmetry of minimizers to the Schrödinger–Poisson–Slater (S–P–S) energy:

$$\underset{\genfrac{}{}{0.0pt}{}{u\in H^1\left({ℝ}^3\right)}{\parallel u{\parallel }_{L^2\left({ℝ}^3\right)}=\rho }}{\inf }\frac{1}{2}\underset{{ℝ}^3}{\int }{\left|\nabla u\right|}^2+\frac{1}{4}\underset{{ℝ}^3}{\int }\underset{{ℝ}^3}{\int }\frac{{\left|u\right(x\left)\right|}^2{\left|u\right(y\left)\right|}^2}{|x-y|}dxdy-\frac{1}{p}\underset{{ℝ}^3}{\int }{\left|u\right|}^{p}dx$$

provided that $2<p<3$ and ρ is small. The main result shows that minimizers are radially symmetric modulo suitable translation.

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