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

Georgiev, Vladimir; Prinari, Francesca; Visciglia, Nicola

Georgiev, Vladimir ; Prinari, Francesca ; Visciglia, Nicola

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 369-376 / Harvested from Numdam

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Résumé
Abstract

On montre la radialité des minimiseurs de lʼénergie de Schrödinger–Poisson–Slater $\inf_{\binom{u \in H^{1} (ℝ^{3})}{∥ u ∥_{L^{2} (ℝ^{3})} = ρ}} \frac{1}{2} \int_{ℝ^{3}} {| \nabla u |}^{2} + \frac{1}{4} \int_{ℝ^{3}} \int_{ℝ^{3}} \frac{{| u (x) |}^{2} {| u (y) |}^{2}}{| x - y |} d x d y - \frac{1}{p} \int_{ℝ^{3}} {| u |}^{p} d x$ 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: $\inf_{\binom{u \in H^{1} (ℝ^{3})}{∥ u ∥_{L^{2} (ℝ^{3})} = ρ}} \frac{1}{2} \int_{ℝ^{3}} {| \nabla u |}^{2} + \frac{1}{4} \int_{ℝ^{3}} \int_{ℝ^{3}} \frac{{| u (x) |}^{2} {| u (y) |}^{2}}{| x - y |} d x d y - \frac{1}{p} \int_{ℝ^{3}} {| u |}^{p} d x$ provided that $2 < p < 3$ and ρ is small. The main result shows that minimizers are radially symmetric modulo suitable translation.

Publié le : 2012-01-01

Zbl 1260.35204

DOI : https://doi.org/10.1016/j.anihpc.2011.12.001

@article{AIHPC_2012__29_3_369_0,
     author = {Georgiev, Vladimir and Prinari, Francesca and Visciglia, Nicola},
     title = {On the radiality of constrained minimizers to the Schr\"odinger--Poisson--Slater energy},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {369-376},
     doi = {10.1016/j.anihpc.2011.12.001},
     zbl = {1260.35204},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_3_369_0}
}

Georgiev, Vladimir; Prinari, Francesca; Visciglia, Nicola. On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energy. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 369-376. doi : 10.1016/j.anihpc.2011.12.001. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_3_369_0/

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