Ground states for semi-relativistic Schrödinger-Poisson-Slater energies

2011/03/14 by Bellazzini, Jacopo, Ozawa, Tohru, Visciglia, Nicola
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1103.2649

Abstract

We prove the existence of ground states for the semi-relativistic Schrödinger-Poisson-Slater energy I^α,β(ρ)=inf_\substacku∈ H^\frac 12(\R³) ∫_\R³|u|² dx=ρ (1)/(2)‖u‖²_{H^\frac 12(\R³)} +α∫∫_\R³×\R³ \frac| u(x)|²|u(y)|²|x-y|dxdy-β∫_\R³|u|^(8)/(3)dx α,β>0 and ρ>0 is small enough. The minimization problem is L² critical and in order to characterize of the values α, β>0 such that I^{α, β}(ρ)>-∞ for every ρ>0, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant S>0 such that (1)/(S)\frac‖φ‖_{L^\frac 83(\R³)}‖φ‖ _{H^\frac 12(\R³)}^\frac 12≤ (∫∫_{\R³× \R³} (|φ(x)|²|φ(y)|²)/(|x-y|)dxdy)^\frac 18 for all φ∈ C^∞₀(\R³). Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm ‖u‖²_{H^\frac 12(\R³)} by the homogeneous one ‖u‖ _{H^\frac 12(\R³)}.

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