### Abstract

We prove the existence of ground states for the semi-relativistic Schrödinger- Poisson-Slater energy (Formula presented) α, β > 0 and ρ > 0 is small enough. The minimization problem is L^{2} critical and in order to characterize 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 (Formula presented) for all φ ∈ C_{0} ^{∞} (R^{3}). Besides, we show that similar compactness property fails if we replace the inhomogeneous Sobolev norm ║u║^{2} _{H1/2(R3)} by the homogeneous one ║u║_{Ḣ1/2(R3)} in the energy above.

Original language | English |
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Pages (from-to) | 353-369 |

Number of pages | 17 |

Journal | Funkcialaj Ekvacioj |

Volume | 60 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

### Keywords

- Concentration-compactness
- Ground states
- Semi-relativistic Schrödinger equation

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

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## Cite this

*Funkcialaj Ekvacioj*,

*60*(3), 353-369. https://doi.org/10.1619/fesi.60.353