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

Vladimir Georgiev; Francesca Prinari; Nicola Visciglia

doi:10.1016/j.anihpc.2011.12.001

JournalsaihpcVol. 29, No. 3pp. 369–376

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

Vladimir Georgiev
Università di Pisa, Dipartimento di Matematica, Largo B. Pontecorvo 5, 56124, Pisa, Italy
Francesca Prinari
Università di Ferrara, Dipartimento di Matematica, via Machiavelli 35, 44100, Ferrara, Italy
Nicola Visciglia
Università di Pisa, Dipartimento di Matematica, Largo B. Pontecorvo 5, 56124, Pisa, Italy

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Abstract

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

u \in H^{1} (R^{3}) ∥ u ∥_{L^{2} (R^{3})} = ρ in f \frac{1}{2} R^{3} \int ∣ \nabla u ∣^{2} + \frac{1}{4} R^{3} \int R^{3} \int \frac{∣ u ( x ) ∣ ^{2} ∣ u ( y ) ∣ ^{2}}{∣ x - y ∣} d x d y - \frac{1}{p} R^{3} \int ∣ 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.

Résumé

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

u \in H^{1} (R^{3}) ∥ u ∥_{L^{2} (R^{3})} = ρ in f \frac{1}{2} R^{3} \int ∣ \nabla u ∣^{2} + \frac{1}{4} R^{3} \int R^{3} \int \frac{∣ u ( x ) ∣ ^{2} ∣ u ( y ) ∣ ^{2}}{∣ x - y ∣} d x d y - \frac{1}{p} R^{3} \int ∣ u ∣^{p} d x

pourvu que $2 < p < 3$ et $ρ$ est petit.

Cite this article

Vladimir Georgiev, Francesca Prinari, Nicola Visciglia, On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 3, pp. 369–376

DOI 10.1016/J.ANIHPC.2011.12.001