## Abstract

We prove and give numerical results for two lower bounds and eleven upper bounds to the optimal constant k_{0} = k_{0}(n,α) in the inequality ∥u∥_{2n/(n-2α)} ≤ k_{0} ∥∇_{u}∥^{α}_{2} ∥u∥^{1-α} _{2}, u ∈ H^{1}(ℝ^{n}), for n = 1, 0 < α ≤ 1/2, and n ≥ 2, 0 < α < 1. This constant k_{0} is the reciprocal of the infimum λ_{n,α} for u ∈ H^{1}(ℝ^{n}) of the functional Λ_{n,α} = ∥∇_{u}∥^{α}_{2} ∥u∥^{1-α} _{2}/∥u∥_{2n/(n-2α)}, u ∈ H^{1}(ℝ^{n}), where for n = 1, 0 < α ≤ 1/2, and for n ≥ 2, 0 < α < 1. The lowest point in the point spectrum of the Schrödinger operator τ = -Δ+q on ℝ^{n} with the real-valued potential q can be expressed in λ_{n,α} for all q _ = max(0,-q) ∈ L^{p}(ℝ^{n}), for n = 1, 1 ≤ p < ∞, and n ≥ 2, n/2 < p < ∞, and the norm ∥q _ ∥p.

Original language | English |
---|---|

Pages (from-to) | 753-778 |

Number of pages | 26 |

Journal | Journal of Mathematical Inequalities |

Volume | 13 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- Lower bound
- Optimal constant
- Sobolev inequality
- Upper bound