AI 中文总结
该论文研究多极 Lp-Hardy 不等式，通过基态表示和变分分析，在临界与次临界区域分类势能，证明阈值下紧性、极值元存在及算子临界性，并揭示系数随 p 变化的行为。
AI 中文摘要
我们研究了 1 < p < n 时一族真正多极 Lp-Hardy 权重的基态、临界性与变分紧性之间的关系。一个在极点处有效的基态表示给出了完整势能在临界与次临界参数区域中的完全分类，并区分了有限能量可达性与不可达的临界性。移除一个几何相互作用项导致第二个变分问题，其最优系数由极点和无穷远处的集中控制。我们证明，在所得阈值之下存在严格间隙时，归一化极小化序列具有强紧性，存在正极值元，并且最优重缩放算子是临界的。对于 1 < p < 2，该间隙在整个完整势能临界集中成立，且最优截断系数严格小于 1；对于 p > 2，在完整势能可达区域中该系数严格大于 1。在二次可达区域内部固定参数处，当 p 趋于 2 时最优值收敛到 1，两侧均具有可达性和临界性。证明结合了零序列刚性和严格极点容量与底层向量场零集上的局部加权紧性。
英文摘要
We study the relation between ground states, criticality, and variational compactness for a family of genuinely multipolar Lp-Hardy weights, 1 < p < n. A ground-state representation valid across the poles gives a complete classification of the full potential into critical and subcritical parameter regions and distinguishes finite-energy attainment from non-attained criticality. Removing a geometric interaction term leads to a second variational problem, whose optimal coefficient is controlled by concentration at the poles and at infinity. We prove that a strict gap below the resulting threshold yields strong compactness of normalized minimizing sequences, a positive extremizer, and criticality of the optimally rescaled operator. For 1 < p < 2, this gap holds throughout the full-potential critical set and the optimal truncated coefficient is strictly below 1; for p >2, the coefficient is strictly above 1 in the full-potential attainment region. At fixed parameters strictly inside the quadratic attained region, the optimal value converges to 1 as p tending to 2, with attainment and criticality on both sides. The proofs combine null sequence rigidity and strict-pole capacity with local weighted compactness across the zero set of the underlying vector field.
